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export mpl...

vv 14242
November 01 2023
3 1

Export the document as Maple code (.mpl).
Then open the .mpl. In Maple 2023, it will appear in a new Code Edit window.
Edit the code, i.e. replace `–` with -

There are not other errors (see the Diagnostics tab), so you can copy & paste the code in the worksheet and execute.

P.S. I use 2D input strictly for presentation purposes (and almost never the Document mode).

degree...

vv 14242
October 25 2023
3 0

ex:=x*y*z^2*w^3:
degree(ex);
 #                              7
factors(ex)[2];
 #               [[y, 1], [z, 2], [w, 3], [x, 1]]

Procedure for an enhanced version...

vv 14242
October 01 2023
3 6 
Miny:=proc(eqs::list, m::list(posint), R:=1000)
description "minimize y, for integers  x,y  satisfying 0<x<y, eqs[1] = 0 mod m[1], ...";
local p,q,X,Y,u,k, ir, Xmin,Ymin,pmin,qmin;
if nops(eqs)<>nops(m) then error "Lists must have same size" fi;
if nops(indets(eqs)) <> 2 then error "Must have 2 variables" fi; 
eval( isolve({(eqs =~ m*~[seq(u[k],k=1..nops(m))])[]}), [_Z1=p,_Z2=q] ):
(X,Y):= (eval([indets(eqs)[]],%)[]);
ir:=IntegerRelations:-LLL([[coeff(X,p),coeff(Y,p)], [coeff(X,q),coeff(Y,q)]]):
(X,Y):= (ir[1,1]*p+ir[2,1]*q, ir[1,2]*p+ir[2,2]*q);
(Xmin,Ymin,pmin,qmin):=(infinity$4):
for p from -R to R do
for q from -R to R do
  if eval(X<1 or X >= Y) then next fi;
  if eval(Y<Ymin) then (Xmin,Ymin,pmin,qmin):=eval([X,Y,p,q])[] fi
od od:
['X','Y','p','q'] =~ eval([Xmin,Ymin,pmin,qmin]);
end proc:

eqs := [154*x+69*y, 13*x+716*y, 23*x+3059*y]:
m := [7^3, 13^3, 23^3]:
Miny(eqs,m);
              [X = 15893, Y = 18837, p = 0, q = 1]

eqs := [154*x+69*y, 13*x+716*y, 23*x+3059*y, 2295*x + 4522*y, 6479*x + 5396*y]:
m := [7^3, 13^3, 23^3, 17^3,  29^3]:
Miny(eqs,m);
        [X = 186957261, Y = 190687133, p = -12, q = -1]

Solution...

vv 14242
September 30 2023
0 13
> 

restart

Minimize   y   for integers  x,y  satisfying:
  0 < x < y,  154*x+69*y =0 mod  7^3 , 13*x+716*y =0 mod 13^3  , 23*x+3059*y =0 mod  23^3.

> 

 

> 

restart;
> 

eqs := 154*x+69*y, 13*x+716*y, 23*x+3059*y;
m := 7^3, 13^3, 23^3;

154*x+69*y, 13*x+716*y, 23*x+3059*y

  

343, 2197, 12167

 (1)

 

For an "ideal Maple" it should work:

 

 

> 

# Optimization:-Minimize(y, {1<=x, x<=y, eqs =~ m*~(u,v,w)}, assume = integer, depthlimit=100000);
> 

# But the problem it too large!
> 

#################################################################################################
> 

Eqs := 154*x+69*y, 13*x+716*y, x+133*y;
M := 7^3, 13^3, 23^2;

154*x+69*y, 13*x+716*y, x+133*y

  

343, 2197, 529

 (2)
> 

sys:=y =-154 * 1/69*x mod M[1] + M[1]*u,  y = -13/716*x mod M[2] + M[2]*v, y = -1/133*x mod M[3] + M[3]*w;

y = 301*x+343*u, y = 494*x+2197*v, y = 175*x+529*w

 (3)
> 

isolve(sys[1]-sys[2]);

{u = _Z1, v = 122*_Z1+193*_Z2, x = -1387*_Z1-2197*_Z2}

 (4)
> 

eval(%, [_Z1=a,_Z2=b]);

{u = a, v = 122*a+193*b, x = -1387*a-2197*b}

 (5)
> 

e1:=eliminate(%, {u,v});

[{u = a, v = 122*a+193*b}, {x+1387*a+2197*b}]

 (6)
> 

isolve(sys[1]-sys[3]);

{u = _Z1, w = 49*_Z1+126*_Z2, x = 203*_Z1+529*_Z2}

 (7)
> 

eval(%, [_Z1=c,_Z2=d]);

{u = c, w = 49*c+126*d, x = 203*c+529*d}

 (8)
> 

e2:=eliminate(%, {u,w});

[{u = c, w = 49*c+126*d}, {-x+203*c+529*d}]

 (9)
> 

ee:=e1[1] union e2[1], e1[2] union e2[2];

{u = a, u = c, v = 122*a+193*b, w = 49*c+126*d}, {-x+203*c+529*d, x+1387*a+2197*b}

 (10)
> 

EE:=eval(ee, [a=u, c=u]);

{u = u, v = 122*u+193*b, w = 49*u+126*d}, {-x+203*u+529*d, x+1387*u+2197*b}

 (11)
> 

X, X1 := solve~(EE[2],x)[];

-1387*u-2197*b, 203*u+529*d

 (12)
> 

s:=eval(isolve(X=X1),[_Z1=p,_Z2=q,_Z3=r]);

{b = p, d = -1513*p-1590*q, u = 502*p+529*q}

 (13)
> 

X0:=eval(X, s);

-698471*p-733723*q

 (14)
> 

Y0:=rhs(eval(eval(sys[1], s), x=X0));

-210067585*p-220669176*q

 (15)
> 

Optimization:-Minimize(Y0, { X0<=Y0, 1<=X0}, assume=integer, depthlimit=1000 );

[0, [p = 0, q = 0]]

 (16)

Unfortunately Minimize uses hardware integers and it results an overflow!
(This is actually a bug!)

Let's minimize the reduced X0, Y0 problem by brute force:

> 

p:='p':q:='q':
> 

(Xmin,Ymin,pmin,qmin):=(infinity$4):
for p from -1000 to 1000 do
for q from -1000 to 1000 do
  if X0<1 or X0 >= Y0 then next fi;
  if Y0<Ymin then (Xmin,Ymin,pmin,qmin):=(X0,Y0,p,q) fi
od od:
(x,y,'p','q') =~ (Xmin,Ymin,pmin,qmin);

x = 1103, y = 26390, p = 562, q = -535

 (17)
> 

 


 

Download diophant.mw

ExtremePoints for piecewise...

vv 14242
September 21 2023
1 2

 

ExtremePoints is not reliable for piecewice defined functions.

 

> 

restart;
> 

EP:=Student:-Calculus1:-ExtremePoints:
> 

piecewise(-1 <= x and x <= 2, x^2, undefined);

piecewise(-1 <= x and x <= 2, x^2, undefined)

 (1)
> 

EP(%); #OK

[-1, 0, 2]

 (2)
> 

piecewise(-1 < x and x < 2, x^2, undefined);

piecewise(-1 < x and x < 2, x^2, undefined)

 (3)
> 

EP(%); #OK

[0]

 (4)
> 

piecewise(-1 <= x and x <= 2, x, undefined);

piecewise(-1 <= x and x <= 2, x, undefined)

 (5)
> 

EP(%); # expected [1,2]

[2]

 (6)
> 

piecewise(x<1, undefined, x<=2, x*sin(1/x), undefined);

piecewise(x < 1, undefined, x <= 2, x*sin(1/x), undefined)

 (7)
> 

EP(%, 1..2); # expected [1,2]

[]

 (8)
> 

EP(x*sin(1/x), 1..2); # OK

[1, 2]

 (9)
> 

piecewise(x<0, undefined, x=0, 0,  x<=1, x*sin(1/x), undefined);

piecewise(x < 0, undefined, x = 0, 0, x <= 1, x*sin(1/x), undefined)

 (10)
> 

EP(%); # expected [1]

[]

 (11)
> 

piecewise(x<-1, undefined, x<0, x*sin(1/x), x=0, 0, undefined);

piecewise(x < -1, undefined, x < 0, x*sin(1/x), x = 0, 0, undefined)

 (12)
> 

EP(%); # expected [-1]

[]

 (13)
> 

 

 

Bug...

vv 14242
September 19 2023
3 1

It is a bug. For solve too.

Both commands work if V is given as a set, or is omitted.
So, use:
PolynomialSystem(F, {V[]});  or  PolynomialSystem(F, {V[]}, explicit); 
or PolynomialSystem(F);

answer...

vv 14242
September 19 2023
0 3
> 

restart
> 

variables := [t, x, y, z]; MyPsi := proc (t, x, y, z) options operator, arrow; psi(t, x, y, z) end proc; zeta_0 := proc (t, x, y, z) options operator, arrow; Zeta_0(t, x, y, z) end proc; zeta_1 := proc (t, x, y, z) options operator, arrow; Zeta_1(t, x, y, z) end proc; zeta_2 := proc (t, x, y, z) options operator, arrow; Zeta_2(t, x, y, z) end proc; zeta_3 := proc (t, x, y, z) options operator, arrow; Zeta_3(t, x, y, z) end proc

[t, x, y, z]

  

proc (t, x, y, z) options operator, arrow; psi(t, x, y, z) end proc

  

proc (t, x, y, z) options operator, arrow; Zeta_0(t, x, y, z) end proc

  

proc (t, x, y, z) options operator, arrow; Zeta_1(t, x, y, z) end proc

  

proc (t, x, y, z) options operator, arrow; Zeta_2(t, x, y, z) end proc

  

proc (t, x, y, z) options operator, arrow; Zeta_3(t, x, y, z) end proc

 (1)
> 

dzeta_0_dt := diff(zeta_0(t, x, y, z), t); dzeta_1_dt := diff(zeta_1(t, x, y, z), t); dzeta_2_dt := diff(zeta_2(t, x, y, z), t); dzeta_3_dt := diff(zeta_3(t, x, y, z), t); dzeta_0_dx := diff(zeta_0(t, x, y, z), x); dzeta_1_dx := diff(zeta_1(t, x, y, z), x); dzeta_2_dx := diff(zeta_2(t, x, y, z), x); dzeta_3_dx := diff(zeta_3(t, x, y, z), x); dzeta_0_dy := diff(zeta_0(t, x, y, z), y); dzeta_1_dy := diff(zeta_1(t, x, y, z), y); dzeta_2_dy := diff(zeta_2(t, x, y, z), y); dzeta_3_dy := diff(zeta_3(t, x, y, z), y); dzeta_0_dz := diff(zeta_0(t, x, y, z), z); dzeta_1_dz := diff(zeta_1(t, x, y, z), z); dzeta_2_dz := diff(zeta_2(t, x, y, z), z); dzeta_3_dz := diff(zeta_3(t, x, y, z), z)

diff(Zeta_0(t, x, y, z), t)

  

diff(Zeta_1(t, x, y, z), t)

  

diff(Zeta_2(t, x, y, z), t)

  

diff(Zeta_3(t, x, y, z), t)

  

diff(Zeta_0(t, x, y, z), x)

  

diff(Zeta_1(t, x, y, z), x)

  

diff(Zeta_2(t, x, y, z), x)

  

diff(Zeta_3(t, x, y, z), x)

  

diff(Zeta_0(t, x, y, z), y)

  

diff(Zeta_1(t, x, y, z), y)

  

diff(Zeta_2(t, x, y, z), y)

  

diff(Zeta_3(t, x, y, z), y)

  

diff(Zeta_0(t, x, y, z), z)

  

diff(Zeta_1(t, x, y, z), z)

  

diff(Zeta_2(t, x, y, z), z)

  

diff(Zeta_3(t, x, y, z), z)

 (2)
> 

eq1 := dzeta_0_dt = MyPsi(t, x, y, z); eq2 := -A^2*dzeta_1_dt+dzeta_0_dx = 0; eq3 := -B^2*dzeta_0_dy+dzeta_0_dy = 0; eq4 := -C^2*dzeta_3_dt+dzeta_0_dz = 0; eq5 := dzeta_1_dx+(diff(A, t))*zeta_0/A = MyPsi(t, x, y, z); eq6 := A^2*dzeta_1_dy+B^2*dzeta_2_dx = 0; eq7 := A^2*dzeta_1_dz+C^2*dzeta_3_dx = 0; eq8 := dzeta_2_dy+(diff(B, t))*zeta_0/B = MyPsi; eq9 := B^2*dzeta_2_dz+C^2*dzeta_3_dy = 0; eq10 := dzeta_3_dz+(diff(C, t))*zeta_0/C = MyPsi(t, x, y, z)

diff(Zeta_0(t, x, y, z), t) = psi(t, x, y, z)

  

-A^2*(diff(Zeta_1(t, x, y, z), t))+diff(Zeta_0(t, x, y, z), x) = 0

  

-B^2*(diff(Zeta_0(t, x, y, z), y))+diff(Zeta_0(t, x, y, z), y) = 0

  

-C^2*(diff(Zeta_3(t, x, y, z), t))+diff(Zeta_0(t, x, y, z), z) = 0

  

diff(Zeta_1(t, x, y, z), x) = psi(t, x, y, z)

  

A^2*(diff(Zeta_1(t, x, y, z), y))+B^2*(diff(Zeta_2(t, x, y, z), x)) = 0

  

A^2*(diff(Zeta_1(t, x, y, z), z))+C^2*(diff(Zeta_3(t, x, y, z), x)) = 0

  

diff(Zeta_2(t, x, y, z), y) = MyPsi

  

B^2*(diff(Zeta_2(t, x, y, z), z))+C^2*(diff(Zeta_3(t, x, y, z), y)) = 0

  

diff(Zeta_3(t, x, y, z), z) = psi(t, x, y, z)

 (3)
> 

dependent_variables := [Zeta_0(t, x, y, z), Zeta_1(t, x, y, z), Zeta_2(t, x, y, z), Zeta_3(t, x, y, z), A(t), B(t), C(t), psi(t, x, y, z)]; independent_variables := [t, x, y, z]

[Zeta_0(t, x, y, z), Zeta_1(t, x, y, z), Zeta_2(t, x, y, z), Zeta_3(t, x, y, z), A(t), B(t), C(t), psi(t, x, y, z)]

  

[t, x, y, z]

 (4)
> 

solutions := pdsolve({eq1, eq10, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9}, dependent_variables);

{Zeta_0(t, x, y, z) = (1/2)*_C1*t^2+(1/2)*(2*_C2*x+2*_C3*z+2*_C4)*t+(1/2)*(A^2*x^2+C^2*z^2)*_C1+A^2*_C10*x+C^2*_C6*z+_C12, Zeta_1(t, x, y, z) = (1/2)*((_C2*x^2+(2*_C1*t+2*_C3*z+2*_C4)*x+2*_C10*t+2*_C11)*A^2+(-C^2*z^2+t^2)*_C2-2*C^2*_C7*z-2*B^2*_C9*y)/A^2, Zeta_2(t, x, y, z) = MyPsi*y-C^2*_C5*z/B^2+_C9*x+_F2(t), Zeta_3(t, x, y, z) = (1/2)*((_C3*z^2+(2*_C1*t+2*_C2*x+2*_C4)*z+2*_C5*y+2*_C6*t+2*_C7*x+2*_C8)*C^2+(-A^2*x^2+t^2)*_C3)/C^2, psi(t, x, y, z) = _C1*t+_C2*x+_C3*z+_C4}

 (5)
> 

for i to nops(solutions) do print(solutions[i]) end do

Zeta_0(t, x, y, z) = (1/2)*_C1*t^2+(1/2)*(2*_C2*x+2*_C3*z+2*_C4)*t+(1/2)*(A^2*x^2+C^2*z^2)*_C1+A^2*_C10*x+C^2*_C6*z+_C12

  

Zeta_1(t, x, y, z) = (1/2)*((_C2*x^2+(2*_C1*t+2*_C3*z+2*_C4)*x+2*_C10*t+2*_C11)*A^2+(-C^2*z^2+t^2)*_C2-2*C^2*_C7*z-2*B^2*_C9*y)/A^2

  

Zeta_2(t, x, y, z) = MyPsi*y-C^2*_C5*z/B^2+_C9*x+_F2(t)

  

Zeta_3(t, x, y, z) = (1/2)*((_C3*z^2+(2*_C1*t+2*_C2*x+2*_C4)*z+2*_C5*y+2*_C6*t+2*_C7*x+2*_C8)*C^2+(-A^2*x^2+t^2)*_C3)/C^2

  

psi(t, x, y, z) = _C1*t+_C2*x+_C3*z+_C4

 (6)

NULL

Download code_solve_system_of_equations_vv.mw

answer...

vv 14242
September 18 2023
2 0 
A219954list := nmax -> 
[seq[scan=`+`](ifelse(n=1, 0, 3^numboccur(convert(n-1, 'base', 2), 1) - ifelse(2^ilog2(n)=n, n/2, 0)), n = 1..nmax)]:

A219954list(33);
 [0, 2, 5, 12, 15, 24, 33, 56, 59, 68, 77, 104, 113, 140, 167,  240, 243, 252,
261, 288, 297, 324, 351, 432, 441, 468, 495,   576, 603, 684, 765, 992, 995]

Do not use floats...

vv 14242
September 12 2023
1 1

NULL

> 

restart
> 

v := 203/100; x0 := 0; y0 := 0; mu := 1; A_coef := (1/2)*(3*(v-2)); Coef_inside_sqrt := (5*(1/10))*sqrt((v-2)/(2*mu*v)); u_exact := proc (t, X, Y) options operator, arrow; A_coef*sech(Coef_inside_sqrt*(X+Y-v*t-x0-y0))^2 end proc

203/100

  

0

  

0

  

1

  

9/200

  

(1/812)*1218^(1/2)

  

proc (t, X, Y) options operator, arrow; A_coef*sech(Coef_inside_sqrt*(X+Y-v*t-x0-y0))^2 end proc

 (1)

1

  

0.45000000e-1

  

0.4298011912e-1

  

proc (t, X, Y) options operator, arrow; A_coef*sech(Coef_inside_sqrt*(X+Y-v*t-x0-y0))^2 end proc

 (2)
> 

u_exact(t,X,Y);

(9/200)*sech((1/812)*1218^(1/2)*(X+Y-(203/100)*t))^2

 (3)
> 

PDE := diff(u(t, x, y), t)+diff(u(t, x, y), x)+diff(u(t, x, y), y)+u(t, x, y)*(diff(u(t, x, y), x))+u(t, x, y)*(diff(u(t, x, y), y))-mu*(diff(u(t, x, y), x, x, t))-mu*(diff(u(t, x, y), y, y, t))

diff(u(t, x, y), t)+diff(u(t, x, y), x)+diff(u(t, x, y), y)+u(t, x, y)*(diff(u(t, x, y), x))+u(t, x, y)*(diff(u(t, x, y), y))-(diff(diff(diff(u(t, x, y), t), x), x))-(diff(diff(diff(u(t, x, y), t), y), y))

 (4)
> 

eval(PDE, u=u_exact)

(27/8120000)*sech((1/812)*1218^(1/2)*(x+y-(203/100)*t))^2*1218^(1/2)*tanh((1/812)*1218^(1/2)*(x+y-(203/100)*t))-(81/8120000)*sech((1/812)*1218^(1/2)*(x+y-(203/100)*t))^4*1218^(1/2)*tanh((1/812)*1218^(1/2)*(x+y-(203/100)*t))-(27/8120000)*sech((1/812)*1218^(1/2)*(x+y-(203/100)*t))^2*tanh((1/812)*1218^(1/2)*(x+y-(203/100)*t))^3*1218^(1/2)+(27/4060000)*sech((1/812)*1218^(1/2)*(x+y-(203/100)*t))^2*tanh((1/812)*1218^(1/2)*(x+y-(203/100)*t))*1218^(1/2)*(1-tanh((1/812)*1218^(1/2)*(x+y-(203/100)*t))^2)

 (5)
> 

simplify(%);

0

 (6)
> 

 

NULL


Download solution_pde_check-exact.mw

global...

vv 14242
September 05 2023
2 0

Technically, you must use:

U,S,Vt := SingularValues(A, output=[':-U',':-S',':-Vt']);

(Of course, in your example, this does not make much sense because U, S, Vt are not needed/used.)

A solution...

vv 14242
August 23 2023
4 4 
restart;
a:=floor(33*sqrt(3)):
R2:=33^2*3:  # the radius^2
L2:=R2*16/6: # the edge^2
nr:=0: W:=Array(1..0):
for  x from 0 to a do
for  y from x to a do
for  z from y to a do
if x^2+y^2+z^2=R2 then nr++: W(nr):=[x,y,z] fi
od od od:
f:=u -> ([u[1],u[2],u[3]], [-u[1],u[2],u[3]],[u[1],-u[2],u[3]], [u[1],u[2],-u[3]],
        [-u[1],-u[2],u[3]], [-u[1],u[2],-u[3]],[u[1],-u[2],-u[3]], [-u[1],-u[2],-u[3]]):
P:=f~([seq](W)): # the integer points on the sphere
n:=nops(P);
#                            n := 96

d2 :=(u,v) -> inner(u-v,u-v):
A:=Matrix( n, (i,j)-> `if`(d2(P[i],P[j])=L2, 1, 0) ):
with(GraphTheory):
G:=Graph(A):
K4 := CompleteGraph(4):
IsSubgraphIsomorphic(K4,G, isomorphism);
#             true, {1 = 95, 2 = 94, 3 = 93, 4 = 89}

T := P[[rhs~(%[2])[]]]; # A desired tetrahedron
# T := [ [33, 33, 33], [-33, -33, 33], [-33, 33, -33], [33, -33, -33]  ]

plots:-display(plottools:-tetrahedron(T), plottools:-sphere([0, 0, 0], sqrt(R2), transparency=0.8), color="Blue");

DirectSearch...

vv 14242
August 12 2023
4 7

The floor method cannot work because NLPSolve finds only local extrema and almost any point is a local max,
actually any point in ( (-5, oo) \ Z )^9.

The DirectSearch package (in Application Center) works, but the solution is of course not guaranteed.

GlobalOptima(add(x[k],k=1..9), 
   [seq(x[k]::integer, k=1..9),seq(x[k]<=x[k+1], k=1..8), x[1]>=-5, x[9]<=50, add(x[k]^3,k=1..9)=0],
   evaluationlimit=50000, maximize);

     [12., [x[1] = -4, x[2] = 2, x[3] = 2, x[4] = 2, x[5] = 2, x[6] = 2, x[7] = 2, x[8] = 2, x[9] = 2], 2169]

Hermite...

vv 14242
August 12 2023
0 0

No, such polynomial would have been found by solve.
(You have a Hermite interpolation problem here!)

u,v as exp...

vv 14242
July 28 2023
2 1 
ex:=F(u^(1/n), v^(1/n))^n - F(u, v):
ex1:=eval(ex, [u=exp(-A), v=exp(-B)]):
expand(simplify(ex1)) assuming positive;  # 0

 

Just use ReducedRowEchelon...

vv 14242
July 19 2023
3 2

Just use ReducedRowEchelonForm and append the identity matrix to your square matrix.

 

> 

restart;
> 

with(LinearAlgebra):
> 

n:=3:
> 

A:=RandomMatrix(n);

Matrix(3, 3, {(1, 1) = 27, (1, 2) = 99, (1, 3) = 92, (2, 1) = 8, (2, 2) = 29, (2, 3) = -31, (3, 1) = 69, (3, 2) = 44, (3, 3) = 67})

 (1)
> 

LinearAlgebra:-ReducedRowEchelonForm(<A|IdentityMatrix(n)>);

Matrix(3, 6, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (1, 4) = -3307/327244, (1, 5) = 2585/327244, (1, 6) = 5737/327244, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 2675/327244, (2, 5) = 4539/327244, (2, 6) = -1573/327244, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1, (3, 4) = 1649/327244, (3, 5) = -5643/327244, (3, 6) = 9/327244})

 (2)
> 

B:=%[..,n+1..];

Matrix(3, 3, {(1, 1) = -3307/327244, (1, 2) = 2585/327244, (1, 3) = 5737/327244, (2, 1) = 2675/327244, (2, 2) = 4539/327244, (2, 3) = -1573/327244, (3, 1) = 1649/327244, (3, 2) = -5643/327244, (3, 3) = 9/327244})

 (3)
> 

A.B; #check
 

Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})

 (4)
> 

 

 

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