## 1540 Reputation

19 years, 141 days

## Use PolyhedralSets....

Try this:

with(PolyhedralSets):
V := [x, y, z];
p1 := x+z-1;
H1 := PolyhedralSet([p1 <= 0], V):
H2 := PolyhedralSet([p1 = 0], V):
H3 := PolyhedralSet([p1 >= 0], V):
C := PolyhedralSet([seq(n >= 0, n = V), seq(n <= 1, n = V)], V):
S := seq(h intersect C, h = [H1, H2, H3]);

ev:= edgeoptions = [color = "black", thickness=0], vertexoptions = [color = "red", symbolsize = 1]:
plots[display](
Plot(S[1], faceoptions = [color = pink, glossiness = 0.25, transparency=0.33], ev),
Plot(S[2], faceoptions = [color = aquamarine, glossiness = 0.25, transparency=0], ev),
Plot(S[3], faceoptions = [color = turquoise, glossiness = 0.25, transparency=0.33], ev),
view=[(-1..2)\$3], scaling = constrained, lightmodel = light3, axes = boxed, insequence=false
);

.

## see ?interp...

y = interp(op(map2(map2, op, [1,2], [[-1, -4], [1, 0], [2, 0], [3, -4]])), x);

## Without floats....

Converting into fractions we get:

Sum(binomial(196, j)*(1/2)^(321/10+j)*(Sum(binomial(109, l)*(-1)^(j+l)*(Sum((1/2)^((31/10)*h)*GAMMA(-41/10+h)*GAMMA(382/41+(10/41)*l+(10/41)*j+(31/41)*h)/(factorial(h)*(321/31+(10/31)*j+h)*GAMMA((10/41)*j+(10/41)*l+6321/410+(31/41)*h)), h = 0 .. infinity)), l = 0 .. 109)), j = 0 .. 196);

## A guess....

I do not really understand what you are saying so my question follows:

We can indentify any point P using: P = [x, y, c]; where c is the color and (x,y) is the coordinate in the plane.
We have a rule: For all ( P[1] = [x[1], y[1], c[1]], P[2] = [x[2], y[2], c[2]] ) (c[1]=c[2]) implies not (y[1]=y[2]).
An observation: for every set of points satisfying above rule, the cardinality of (subset of points) having only particular (y, c) coordinates is not greater than 1.

You've given us a set of 63 points, call it set S. Which are you trying to do:
1) choose a subset of S that satisifies the rule
2) partition S into subsets satisfying the rule

[{x = 4, x = 5, x = 6, x = 7, x = 2, x = 3, x = 1}, {y = 1, c = 1}], [{x = 4, x = 5, x = 6, x = 7, x = 2, x = 3, x = 1}, {c = 2, y = 2}], [{x = 4, x = 5, x = 7, x = 1}, {c = 3, y = 3}], [{x = 4, x = 5, x = 7, x = 1}, {y = 4, c = 3}], [{x = 6, x = 2, x = 3}, {c = 4, y = 3}], [{x = 6, x = 2, x = 3}, {y = 4, c = 4}], [{x = 5, x = 7, x = 3, x = 1}, {c = 5, y = 5}], [{x = 5, x = 7, x = 3, x = 1}, {y = 6, c = 5}], [{x = 4, x = 6, x = 2}, {c = 6, y = 6}], [{x = 4, x = 6, x = 2}, {c = 6, y = 5}], [{x = 7, x = 2, x = 1}, {c = 7, y = 9}], [{x = 7, x = 2, x = 1}, {c = 7, y = 7}], [{x = 7, x = 2, x = 1}, {c = 7, y = 8}], [{x = 5, x = 3}, {c = 8, y = 7}], [{x = 5, x = 3}, {c = 8, y = 8}], [{x = 5, x = 3}, {c = 8, y = 9}], [{x = 4, x = 6}, {c = 9, y = 8}], [{x = 4, x = 6}, {c = 9, y = 7}], [{x = 4, x = 6}, {c = 9, y = 9}];

## Linear transform....

My one bet is the OP is looking for this:

X := x, y;
V := x^4, x^3*y, x^2*y^2, x*y^3, y^4;
P := (-x+sqrt(3)*y)/2,(-sqrt(3)*x-y)/2;
VP := expand(unapply([V], X)(P));
map(proc(E) local C, W; C := coeffs(E, {x, y}, 'W'); subs( [W] =~ [C], ({V} minus {W}) =~ 0,  [V]) end, VP);

## No need for recursion....

You can simply solve for J and use that, no need for recursion.

J=(n->(2*(n-2^(ilog[2](n)))+1));

Can you prove the formula?

## Use diff and not D....

I would suspect the reason is you are not using diff and mtaylor does. So try this code:

restart;
f := (x,y) -> 1/(2 + x*y^2):
P := proc(x,y,x0,y0,N) local p,q; options operator, arrow;  add(1/factorial(n)*add(binomial(n,k)*subs(p=x-x0,q=y-y0,diff(f(p,q),[p\$(n-k),q\$k])*p^(n-k)*q^k), k=0..n), n=0..N): end;
P(x,y,x0,y0,30):

## Exp....

Maple 5.4 says:

Int(ln(2^(1/2)*t^(1/2))*exp(-t),t = 0 .. infinity) = 1/2*ln(2)-1/2*gamma;

## Combinat package....

This code creates a polynomial in x,y,z of degree 8:

f := proc(V, d) local n; map(mul, combinat[choose]([1\$d, op(map(`\$`, V, d))], d)) end;
L := f([x, y, z], 8);
add(c[i] * L[i], i = 1..nops(L));

Modify as needed.

## Surd....

You're obviousy thinking of surd. Look at the following:
surd(x,3) assuming x, real;

it will give you:
signum(x)*abs(x)^(1/3);

## With unapply....

Try this code:

restart;
rec:=collect(solve({2*(k+1)*X(k+1)+(k+1)*Y(k+1)-X(k)-Y(k)+(1)/k!,(k+1)*X(k+1)+(k+1)*Y(k+1)+2*X(k)+Y(k)+(1)/k!},{X,Y}(k+1)), [X,Y]);
eq:=map2((expand@lhs = proc(f) setattribute(f, remember) end@rhs)@map, unapply, subs(k = k-1, rec), k);
init:=X(0) = 2, Y(0) = 1;
assign(eq, init);
L:=seq([X, Y](i),i=0..20);

## Use content....

((x->x), (content@numer)/(content@denom))((1/1296)*cBooP0-(1/1296)*cSRP0-(1/1296)*tStartRamp*f__SR/N);
proc(x,y) ``(x)*y end((%[1]/%[2]),%[2]);

This is for printing purposes only. Expansion will always occur when multiplying by a rational. More options in ?content

## "simple" formula....

I think this is the plot of your function (Int(1/sqrt(sin(x0)-sin(x)),x=0..x0)):

plot([arcsin(s), 2*EllipticF(sqrt(s)/sqrt(s+1), I*sqrt(-s^2+1)/(-1+s))/sqrt(1-s), s = 0 .. 1], view = [0 .. Pi/2, 0 .. 5]);

#and another one (edit):

plot([arcsin(2*t^2-1), sqrt(2)*(EllipticK(t)-EllipticF(sqrt(2)/(2*t), t)), t = sqrt(2)/2 .. 1], view = [0 .. Pi/2, 0 .. 5]);

Perhaps you can use this definition to invert your function.

## Coeffs was intended for this:...

proc(P, V, f) local l; zip(f, [coeffs](P, V, 'l'), [l]) end(14*c^4 + 84*c^3*d + 180*c^2*d^2 + 165*c*d^3 + 55*d^4 + 5*c^3 + 21*c^2*d + 28*c*d^2 + 12*d^3 + 2*c^2 + 5*c*d + 3*d^2 + c + d + 1, [c,d], (p,q) -> `if`(degree(q, [c, d])=3, [p, q], NULL));

## A partial solution....

This is not a complete solution but you will find it useful:

 A := cos(2*Pi*(x+y-2*z))+cos(2*Pi*(y+z-2*x))+cos(2*Pi*(z+x-2*y)) = 0; B := {0 <= x, x <= 1, y <= 1, z <= 1, x < y, y < z}; V := [op](map2(op, 1, indets(A, trig))); E0 := [n || (\$1..3)] =~ V; E := eliminate(E0, {x, y, z, n1}); A2 := add(cos(n), n = factor(subs(E[1], V))); S := subs(E[1], B); solve(S, {n2, n3}); solve(A2, {n2, n3}); #solve(A2, {n2, n3}, allsolutions=true);

Thumb if you like.

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