How to Use vTool.mplThe file vTool.mpl contains maple functions to compute volumes and curvatures of bitnets.First the file must be read into the current session:read "c:/cygwin/home/cr569/vTool.mpl";A short description of most of the functions defined by the module can be obtained with:showVolsTool();To define a new dag use the function Dag. Enter the number of nodes and the links as in the following example that shows the complete dag on 3 nodes:dag := Dag(3,[1,2,1,3,2,3]);The exact volume of this dag model is obtained with,VolExact(dag);This is half of the volume of the unit sphere in dimension 2^3-1 = 7.evalf(%);Lower and upper bound approximations are obtained with:LowVol(dag); HighVol(dag);In the case of complete dags the low and high approximations are exact but that's not always the case.When the exact computation fails one can try an approximation by MCMC with:eps := 0.0001: VolMCMC(dag,eps);evalf(%);Computing the Ricci scalar, the Total curvature and the Mean curvature of a dag:The Scalar Curvature formula is obtained with:SCformula(3,[1,2,1,3,2,3]);In the case of complete dags is constant but in general it returns a formula (often very complicated) that depends on the parameters for the model.To obtain the integral of the Ricci scalar and the mean curvature (i.e. total curvature over the volume of the dag) just:IntRicci(3,[1,2,1,3,2,3]);MeanCurvature(dag);The mean curvature of dags is always observed to be half an integer! If you find a proof or a counter example please let me know (carlos@math.albany.edu) .The Line of Three:If we erase the link from 1 to 3 in the complete dag of 3 nodes we obtain the directed line of 3: 1->2->3. The hypothesis space that this dag represents has dimension 5.dag := Dag(3,[1,2,2,3]);The exact computation of the volume seems to be out of the reach of current Maple!But the function Line(n) returns the volume of the directed line of n nodes.Line(3);evalf(%);The MCMC approximation in this case is:VolMCMC(dag,0.001);evalf(%);LowVol(dag);HighVol(dag);A good guess is obtained by taking the geometric mean of the Low and High approximations as:approxvol := sqrt(LowVol(dag)*HighVol(dag));The formula for the Ricci scalar is no longer constant,R := SCformula(3,[1,2,2,3]);It simplifies considerably as the following computation shows,r := (1-t1)*t2 + t1*t3:Rsimp := 5 - 1/(2*r*(1-r));simplify(Rsimp-R);To compute the total curvature we need to integrate the Ricci scalar w.r.t. the volume element of the dag. The volume element is given by the Jeffreys Prior,dvol := JP(dag);Int(Rsimp*dvol,DofI({1,2,3}));u := (1-x)*y+x*z:ma := Int(1/sqrt(u*(1-u)*y*(1-y)*z*(1-z)),[x=(0..1),y=(0..1),z=(0..1)],method=_MonteCarlo,epsilon=0.001);evalf(%);This is in fact evalf(Pi^3);TotR := 5*Line(3) - Pi^2/2*Pi^3;TotR/Line(3);TTdSMApJNVJUQUJMRV9TQVZFLzE4ODI4OTc2WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiFGJ0YnIiIiRidGJ0YoRihGJ0YmCg==TTdSMApJNVJUQUJMRV9TQVZFLzIwODQ2OTQwWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiFGJ0YnIiIiRidGJ0YnRihGJ0YmCg==