Recently here Mapleprimes member @Icz asked about generating all minimal edge cuts of a graph. I gave a brute force algorithm based on testng all bipartitions of the vertices. I then looked into improving the method by using fundamental cutsets to generate the cutsets. A description of the method and the ideas behind it is given here, together with a comparison of the two methods.

[Edit - see below for a newer version.]

Download CutSets.mw

L:=[["O",3.85090000,0.45160000,0.00120000],
["O",−2.59990000,1.40410000,−0.00180000],
["N",−1.57050000,−0.71710000,0.00010000],
["C",−0.20660000,−0.42310000,−0.00020000],
["C",0.22050000,0.90470000,0.00040000],
["C",0.72980000,−1.45700000,−0.00070000],
["C",1.58410000,1.19860000,0.00020000],
["C",2.09330000,−1.16290000,−0.00070000],
["C",2.52040000,0.16480000,−0.00030000],
["C",−2.64850000,0.17820000,0.00090000],
["C",−3.97350000,−0.54200000,0.00100000],
["H",−0.44360000,1.75770000,0.00120000],
["H",0.41130000,−2.49630000,−0.00100000],
["H",−1.80100000,−1.70860000,0.00010000],
["H",1.90530000,2.23700000,0.00090000],
["H",2.81800000,−1.97260000,−0.00080000],
["H",−4.06550000,−1.14630000,−0.90580000],
["H",−4.79040000,0.18440000,0.02880000],
["H",−4.04450000,−1.18860000,0.88020000],
["H",3.96500000,1.41760000,0.00170000]];

S:={{1,9},{1,20},{2,10},{3,4},{3,10},{3,14},{4,5},{4,6},
{5,7},{5,12},{6,8},{6,13},{7,9},{7,15},{8,9},{8,16},{10,11},
{11,17},{11,18},{11,19}}

Now I have list L above a Set S below 

Now we delete from L all having "H" in first position and also  the subsets of the set S which as all contain those  list position number where L had "H"

Kind help with function which can

Takes input of the list and sets returns the new list and set where the above are done 

First of all I would like to wish all of you a happy, prosperous but especially healthy 2023! I have again a beginner question. Why is test2 not working in the attached document?

Thank you so much for your assistance!

QuestionFDS.mw

I remember a section “Tell us what we can do better” at the bottom of online help pages. I used this whenever I came across a potential error worth investigating.

Has this section disappeared (I hope not) or do I have a browser issue?

restart;
with(geometry);
with(plots);
_EnvHorizontalName = 'x';
_EnvVerticalName = 'y';
Vdot := proc(U, V) local i; add(U[i]*V[i], i = 1 .. 2); end proc;
dist := proc(M, N) sqrt(Vdot(M - N, M - N)); end proc;
EQ := proc(M, N) local eq, a, b, c; eq := simplify(expand((y - M[2])/(x - M[1]) - (N[2] - M[2])/(N[1] - M[1]))*(x - P1[1])*(P2[1] - P1[1])); a := coeff(eq, x); b := coeff(eq, y); c := tcoeff(eq, [x, y]); RETURN(-a*x/c - b*y/c - 1); end proc;
R := 5;
ang := [2/3*Pi, -3*Pi*1/4, -Pi*1/6];
seq(point(`||`(P, i), [R*cos(ang[i]), R*sin(ang[i])]), i = 1 .. 3);
seq(dsegment(`||`(seg, i), [`||`(P, i), `||`(P, irem(i, 3) + 1)]), i = 1 .. 3);
circle(cir, [point(OO, [0, 0]), R]);
sol := solve(subs(x = 2, Equation(cir, [x, y])), y);
point(A, [2, sol[1]]);
triangle(Tri, [P1, P2, P3]);
incircle(inc, Tri, 'centername' = oo);
circle(Cr, [A, oo]);
sol := solve({Equation(Cr, [x, y]), Equation(inc, [x, y])}, {x, y});
point(H1, [subs(sol, x), subs(sol, y)]);
line(L, [A, oo]);
reflection(H2, H1, L);
line(L1, [A, H1]);
line(L2, [A, H2]);
Equation(cir, [x, y]);
Equation(L1, [x, y]);
sol := solve({Equation(L1, [x, y]), Equation(cir, [x, y])}, {x, y});
evalf(%);
point(M1, [subs(sol, x), subs(sol, y)]);
sol2 := solve({Equation(L2, [x, y]), Equation(cir, [x, y])}, {x, y});
evalf(%);
point(M2, [subs(sol2, x), subs(sol2, y)]);
triangle(TR, [M1, M2, A]);
display([draw([P1(symbol = solidcircle, symbolsize = 8, color = blue), P2(symbol = solidcircle, symbolsize = 8, color = blue), P3(symbol = solidcircle, symbolsize = 8, color = blue), A(symbol = solidcircle, symbolsize = 8, color = black), H1(symbol = solidcircle, symbolsize = 8, color = black), H2(symbol = solidcircle, symbolsize = 8, color = black), L1(color = black), L2(color = black), seg1(color = magenta), seg2(color = magenta), seg3(color = magenta), Cr(color = black), cir(color = magenta), inc(color = blue)]), textplot([seq([coordinates(`||`(P, i))[], convert(`||`(P, i), string)], i = 1 .. 3)], 'align' = {'above', 'left'})], view = [-6 .. 10, -15 .. 6], scaling = constrained, size = [800, 800], axes = none);
It seems that there is conusion between M1 and M2. How to write letters A, M1 ,M2, H1, H2 ? Thank you.

The range is wrong. For details, see below, please.
 

Download SmartPlots.mw

The help page claims that smartplot(..) will call 2-D plot procedures ultimately, but why is the smartplots command incompatible with the use of assume?

Good day everyone,

I have been having problems with a system of PDE solution using `numeric`.

It's giving me the error code "Error, (in pdsolve/numeric/par_hyp) input system is too far from a 'standard' form (see ?pdsolve,numeric for more detail)" and I have checked to the best of my ability for the error but could not see anything.

The code is attached below.

Please, anyone with useful information should help. Thanks

Numeric.mw

How do I find the solutions "left" and "right" with only answers in the range 0 to +1?

The domain of eq1 and eq2 is 0 <=beta <= 1.

So I'm only interested in left[1] and right[2]:

Here is the code:

eq[1]:=(1/6)*beta^3+(1/2)*xi^2*beta-(1/2)*beta^2-(1/2)*xi^2+(1/3)*beta = 0;

eq[2]:=(1/6)*beta*(beta^2+3*xi^2-6*xi+2) = 0;

left:=solve(eq[1],xi);

right:=solve(eq[2],xi);

left:=plot(left[1],beta=0..1,thickness=2,color=blue);

right:=plot(right[2],beta=0..1,thickness=2,color=blue);

How do I plot a volume of revolution? I can plot other volumes using Student[Calculus1]]:-VolumeOfRevolution, but not this one. I get a blank plot. I do get the correct volume from output=value..

How do I plot this using plot3d?

restart;
a := 0; b := 1;
f := (x) -> x^2+2;
g := (x) -> 1/2*x+1;
V := int(f(x)^2 - g(x)^2,x=a..b)*Pi;
Student[Calculus1]:-VolumeOfRevolution(f(x),g(x),x=a..b,output=value);
Student[Calculus1]:-VolumeOfRevolution(f(x),g(x),x=a..b,output=plot);
 

I cannot plot the **erf** function, please see below, what's wrong ?

Does everyone have a good idea of ​​the work of the Draghilev method? For example, in this answer https://www.mapleprimes.com/questions/235407-The-Second-Example-Of-Finding-All-Solutions#answer291268
( https://www.mapleprimes.com/questions/235407-The-Second-Example-Of-Finding-All-Solutions )
there was a very successful attempt by Rouben Rostamian to calculate the line of intersection of surfaces without applying the Draghilev method.
Let now not 3d, but 8d. And how will the solve command work in this case? Imagine that aij ((i=1..7,j=1..8)) are partial derivatives, and xj (,j=1..8) are derivatives, as in the above example. f8 is responsible for the parametrization condition.
 

restart;
 f1 := a11*x1+a12*x2+a13*x3+a14*x4+a15*x5+a16*x6+a17*x7+a18*x8; 
 f2 := a21*x1+a22*x2+a23*x3+a24*x4+a25*x5+a26*x6+a27*x7+a28*x8; 
 f3 := a31*x1+a32*x2+a33*x3+a34*x4+a35*x5+a36*x6+a37*x7+a38*x8; 
 f4 := a41*x1+a42*x2+a43*x3+a44*x4+a45*x5+a46*x6+a47*x7+a48*x8;
 f5 := a51*x1+a52*x2+a53*x3+a54*x4+a55*x5+a56*x6+a57*x7+a58*x8; 
 f6 := a61*x1+a62*x2+a63*x3+a64*x4+a65*x5+a66*x6+a67*x7+a68*x8; 
 f7 := a71*x1+a72*x2+a73*x3+a74*x4+a75*x5+a76*x6+a77*x7+a78*x8;
 f8 := x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2-1; 
allvalues(solve({f1, f2, f3, f4, f5, f6, f7, f8}, {x1, x2, x3, x4, x5, x6, x7, x8}))

And this is how the Draghilev method works in this case.
 

restart; with(LinearAlgebra):
f1 := a11*x1+a12*x2+a13*x3+a14*x4+a15*x5+a16*x6+a17*x7+a18*x8; 
f2 := a21*x1+a22*x2+a23*x3+a24*x4+a25*x5+a26*x6+a27*x7+a28*x8; 
f3 := a31*x1+a32*x2+a33*x3+a34*x4+a35*x5+a36*x6+a37*x7+a38*x8; 
f4 := a41*x1+a42*x2+a43*x3+a44*x4+a45*x5+a46*x6+a47*x7+a48*x8; 
f5 := a51*x1+a52*x2+a53*x3+a54*x4+a55*x5+a56*x6+a57*x7+a58*x8;
f6 := a61*x1+a62*x2+a63*x3+a64*x4+a65*x5+a66*x6+a67*x7+a68*x8; 
f7 := a71*x1+a72*x2+a73*x3+a74*x4+a75*x5+a76*x6+a77*x7+a78*x8;

n := 7; 
x := seq(eval(cat('x', i)), i = 1 .. n+1):
 F := [seq(eval(cat('f', i)), i = 1 .. n)]: 
A := Matrix(nops(F), nops(F)+1):
 for j to nops(F) do for i to nops(F)+1 do A[j, i] := op(1, op(i, op(j, F))) 
end do:
           end do: 

# b[i] and b[n+1] are solutions of a linear homogeneous system and at the 
# same time they serve as the right-hand sides of an autonomous ODE.
for i to n do

 b[i] := Determinant(DeleteColumn(ColumnOperation(A, [i, n+1]), n+1)) 
                                                                     end do:
 b[n+1] := -Determinant(DeleteColumn(A, n+1)):

Only the original seven linear homogeneous equations with eight variables are needed. We solve them according to Cramer's rule, and in order to have uniqueness when solving the ODE, we use a point on the curve (according to the theory). (This point is sought in any convenient way.)
If we want to get a parameterization, then additionally, directly in dsolve, we can add the following:
 

for i to n do 
b[i] := simplify(Determinant(DeleteColumn(ColumnOperation(A, [i, n+1]), n+1))) 
end do:
b[n+1] := simplify(-Determinant(DeleteColumn(A, n+1))); 
deqs := seq(diff(x[i](s), s) = b[i]/(b[1]^2+b[2]^2+b[3]^2+b[4]^2+b[5]^2+b[6]^2+b[7]^2+b[8]^2)^.5, i = 1 .. n+1):

In an old post, vv reported a bug in simpl/max, which has been "fixed" in Maple 2018. However, it seem that such repairs are not complete enough.
For example, suppose it is required to find the (squared) distance between the origin and a point on x³ - x + y² = ⅓ which is closest to the origin. In other words, one needs to minimize x²＋y² among the points on this curve, i.e., 

extrema(x^2 + y^2, {x^3 + y^2 - x = 1/3}, {x, y}, 's'); # in exact form

Unfortunately, an identical error message appears again: 

Download tryHarder.mws

How about changing the values of parameter ?

for a from -3 by 3/27 to 3 do
    try
        extrema(x^2 + y^2, {x^3 + y^2 - x = a}, {x, y}); 
    catch:
        print(a); 
    end;
od;
                               -1
                               --
                               3 

                               -2
                               --
                               9 

                               -1
                               --
                               9 

                               1
                               -
                               9

                               2
                               -
                               9

                               1
                               -
                               3

By the way, like extrema, Student[MultivariateCalculus]:-LagrangeMultipliers also executes the Lagrange Multiplier method, but strangely, 

Student[MultivariateCalculus][LagrangeMultipliers](y^2 + x^2, [x^3 + y^2 - x - 1/3], [x, y], output = plot):

does not cause any errors.

Suppose I have a matrix representing a magma such as

matrix(n,n, (i,j)->E[i].E[j])

First, if E is a set of integers then . is converged to * for some reason.

Second, is there any way to simplify the presentation of this matrix? I am using subs to simply but I have to essentially hand code every entry which sort of defeats the purpose. Is there any way to present a matrix and then have the resultant simplified as far as possible?

E.g., subs does not allow one to do something like ?.E[3] = E[2] where ? represents any value. E.g., if I have an identity it would be nice to simplify all elements quickly with one expression rather than having one for all possibilities(essentially 2n).

Also, magma's only let positive integers which seems kinda restrictive(why not create map internally to handle it?).

Also, is there any way to use special chars like -? I'm using d but I'd prefer using special chars so it is visually more congruent to what I'm doing. I realize - is also a minus but maybe there is someway around it(unicode?)?

Hello everybody! Happy new year!

Allright, well, the question states the following: "What is the maximum product of the fromula x*y*z which has the sum of 6 if you add x, y, and z together".

Now, i am seeing what they are doing, but somewhere down the line in the explaination they say "(after some calculations)". 

Isnt there a faster way to do it? And what did they do in the example to get to the result?

Thank you very much!

Greetings,

The Function    

Download Mapleprimes_Question_Book_2_Paragraph_5.11_Example_35.mw

I have just installed the Google Maps package but when I run the commands I only get empty results and some errors

Any thoughts of what can be wrong. Have latest Montery OSX 

/Anders