vv

12153 Reputation

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8 years, 3 days

MaplePrimes Activity


These are answers submitted by vv

 

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)

 

 

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);

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

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]

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

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.)

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");

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]

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

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 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)

 

 

You just need a pencil for this.

c=5, a=12/2=6, b^2 = a^2 - c^2 = 11 

So, x^2/36 + y^2/11 = 1.

 

I see the QuantifierElimination package (new in Maple 2023) as experimental.
Here is a much simpler problem and the solution obtained by  QuantifierElimination.

 

restart;

with( QuantifierElimination );

[CylindricalAlgebraicDecompose, DeleteFormula, InsertFormula, PartialCylindricalAlgebraicDecompose, QuantifierEliminate, QuantifierTools]

(1)

f:=(x-1)/(x^2+x+1);

(x-1)/(x^2+x+1)

(2)

fmax:=evala(maximize(f));

-1+(2/3)*3^(1/2)

(3)

forall(x, f <= K);

forall(x, (x-1)/(x^2+x+1) <= K)

(4)

sol:=QuantifierEliminate( % ); ### ?

And(Or(And(-K <= 0, Or(K <> 0, And(2*K <= 1, Or(2*K-1 <> 0, And(-K <= 0, Or(K <> 0, And(K <= 0, Or(K <> 0, -K < 1)))))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(K <> 0, K <= 1, -K^5-2*K^4-4*K^3-6*K^2-3*K < 0, And(K^3+2*K^2+K = 0, K^2+3 <> 0)), Or(K <> 0, -K <= -1, -K^5-2*K^4-4*K^3-6*K^2-3*K < 0, And(K^3+2*K^2+K = 0, K^2+3 <> 0)), Or(K <> 0, K-1 = 0, And(-K^5-2*K^4-4*K^3-6*K^2-3*K <= 0, Or(K^5+2*K^4+4*K^3+6*K^2+3*K <> 0, And(2*K^5+7*K^4+6*K^2-18*K <= -3, Or(2*K^5+7*K^4+6*K^2-18*K+3 <> 0, And(-K^3-3*K^2-5*K <= -1, Or(K^3+3*K^2+5*K-1 <> 0, And(2*K^3+5*K^2-8*K <= -1, Or(2*K^3+5*K^2-8*K+1 <> 0, -K < 0)))))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(-K <= 0, -3*K^2-6*K <= -1, -3*K^3+K^2 < 0, K^3 <> 0), Or(K <= 0, -3*K^2-6*K <= -1, -3*K^3+K^2 < 0, K^3 <> 0), Or(K <= 0, 3*K^2+6*K-1 <> 0, 3*K^2-K <> 0), Or(K <> 0, -K <= -1, And(-K^5-2*K^4-4*K^3-6*K^2-3*K <= 0, Or(K^5+2*K^4+4*K^3+6*K^2+3*K <> 0, And(2*K^5+7*K^4+6*K^2-18*K <= -3, Or(2*K^5+7*K^4+6*K^2-18*K+3 <> 0, And(-K^3-3*K^2-5*K <= -1, Or(K^3+3*K^2+5*K-1 <> 0, And(2*K^3+5*K^2-8*K <= -1, Or(2*K^3+5*K^2-8*K+1 <> 0, -K < 0)))))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(K <= 0, -3*K^2-6*K <= -1, 3*K^3-K^2 < 0, K^3 <> 0), Or(-K <= 0, -3*K^2-6*K <= -1, 3*K^3-K^2 < 0, K^3 <> 0), Or(-K <= 0, 3*K^2+6*K-1 <> 0, 3*K^2-K <> 0), Or(K <= 0, -3*K^2-6*K <= -1, And(Or(3*K^3+6*K^2-K <= 0, 3*K^5+6*K^4-K^3 <= 0), Or(-3*K^2+K < 0, And(3*K^3+6*K^2-K <= 0, -3*K^5-6*K^4+K^3 <= 0)), Or(9*K^5+15*K^4-9*K^3+K^2 < 0, 3*K^5+6*K^4-K^3 <> 0, And(Or(2*K^2+5*K <= 1, K^3+5*K^2+6*K <= 1), 2*K^2+5*K <= 1, -K^3-5*K^2-6*K <= -1, Or(-2*K^2-5*K < -1, K^3+5*K^2+6*K-1 <> 0, And(Or(-K <= 0, 4*K^3+8*K^2-K <= 0), Or(K < 0, And(-K <= 0, -4*K^3-8*K^2+K <= 0)), Or(K^2 < 0, 4*K^3+8*K^2-K <> 0, -K < 0)))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(-K <= 0, -3*K^2-6*K <= -1, And(Or(-3*K^3-6*K^2+K <= 0, -3*K^5-6*K^4+K^3 <= 0), Or(3*K^2-K < 0, And(-3*K^3-6*K^2+K <= 0, 3*K^5+6*K^4-K^3 <= 0)), Or(9*K^5+15*K^4-9*K^3+K^2 < 0, 3*K^5+6*K^4-K^3 <> 0, And(Or(-2*K^2-5*K <= -1, -K^3-5*K^2-6*K <= -1), Or(-2*K^2-5*K < -1, K^3+5*K^2+6*K-1 <> 0, And(Or(K <= 0, -4*K^3-8*K^2+K <= 0), Or(-K < 0, And(K <= 0, 4*K^3+8*K^2-K <= 0)), Or(K^2 < 0, 4*K^3+8*K^2-K <> 0, -K < 0)))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(-K <= 0, -3*K^2-6*K <= -1, And(Or(-3*K^3-6*K^2+K <= 0, -3*K^5-6*K^4+K^3 <= 0), Or(-3*K^2+K < 0, And(-3*K^3-6*K^2+K <= 0, 3*K^5+6*K^4-K^3 <= 0)), Or(-9*K^5-15*K^4+9*K^3-K^2 < 0, 3*K^5+6*K^4-K^3 <> 0, And(Or(-2*K^2-5*K <= -1, -K^3-5*K^2-6*K <= -1), -2*K^2-5*K <= -1, K^3+5*K^2+6*K <= 1, Or(2*K^2+5*K < 1, K^3+5*K^2+6*K-1 <> 0, And(Or(K <= 0, -4*K^3-8*K^2+K <= 0), Or(K < 0, And(K <= 0, 4*K^3+8*K^2-K <= 0)), Or(-K^2 < 0, 4*K^3+8*K^2-K <> 0, -K < 0)))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(K = 0, -3*K^2-6*K < -1, And(Or(3*K^4+6*K^3-K^2 <= 0, 3*K^6+6*K^5-K^4 <= 0), Or(-3*K^3+K^2 < 0, And(3*K^4+6*K^3-K^2 <= 0, -3*K^6-6*K^5+K^4 <= 0)), Or(9*K^5+15*K^4-9*K^3+K^2 < 0, 3*K^5+6*K^4-K^3 <> 0, And(Or(2*K^3+5*K^2-K <= 0, K^4+5*K^3+6*K^2-K <= 0), Or(K < 0, And(2*K^3+5*K^2-K <= 0, -K^4-5*K^3-6*K^2+K <= 0)), Or(-2*K^2-5*K < -1, K^3+5*K^2+6*K-1 <> 0, And(Or(-K^2 <= 0, 4*K^4+8*K^3-K^2 <= 0), Or(K^2 < 0, And(-K^2 <= 0, -4*K^4-8*K^3+K^2 <= 0)), Or(K^2 < 0, 4*K^3+8*K^2-K <> 0, -K < 0)))))), And(K = 0, K-1 = 0, K+1 = 0)), Or(K <= 0, -3*K^2-6*K <= -1, And(Or(3*K^3+6*K^2-K <= 0, 3*K^5+6*K^4-K^3 <= 0), Or(3*K^2-K < 0, And(3*K^3+6*K^2-K <= 0, -3*K^5-6*K^4+K^3 <= 0)), Or(-9*K^5-15*K^4+9*K^3-K^2 < 0, 3*K^5+6*K^4-K^3 <> 0, And(Or(2*K^2+5*K <= 1, K^3+5*K^2+6*K <= 1), Or(2*K^2+5*K < 1, K^3+5*K^2+6*K-1 <> 0, And(Or(-K <= 0, 4*K^3+8*K^2-K <= 0), Or(-K < 0, And(-K <= 0, -4*K^3-8*K^2+K <= 0)), Or(-K^2 < 0, 4*K^3+8*K^2-K <> 0, -K < 0)))))), And(K = 0, K-1 = 0, K+1 = 0)))

(5)

should_be = (K >= fmax);   # !!

should_be = (-1+(2/3)*3^(1/2) <= K)

(6)

is(eval(sol, K=fmax));            # ckeck

is(eval(sol, K=fmax-10^(-20)));   #

true

 

false

(7)

 

N.B. Why do you use such an unusual UserProfile?  The output of   2*x + 30  is:

 

The system is not very simple, but I agree that SolveTools:-SemiAlgebraic is very slow in general, and I had not the patience for a result.

Anyway, the result (for eq1) seems to be (and is confirmed by MMA):

a := RootOf(_Z^3 - 4*_Z^2 - 27, 5.054 .. 5.063):
sol := {z=0, y = x, a < x}, {y=0, z = x, a < x}, {x=0, y = z, a < z};

plots:-spacecurve([[t,t,0], [t,0,t], [0,t,t]], t=a..3*a, color=[red,blue,green], thickness=5);

Can the equation (x-a)^4 + (x-b)^4 = c have four different integer solutions,where  a, b, c  are integers?

 

The answer is NO.  Actually not even four real solutions!

Suppose the polynomial  f := (x-a)^4 + (x-b)^4 - c  has 4 real solutions. After a shift, we may suppose that one of the solutions is 0, so c = a^4 + b^4,  a, b nonzero.

 

restart;

f := (x-a)^4 + (x-b)^4 - a^4 - b^4;

(x-a)^4+(x-b)^4-a^4-b^4

(1)

factor(f);

-2*x*(a+b-x)*(2*a^2-2*a*b-a*x+2*b^2-b*x+x^2)

(2)

 

f1:=op(-1,%);

2*a^2-2*a*b-a*x+2*b^2-b*x+x^2

(3)

f1  must have two real solutions

d1:=discrim(f1,x); # should be >0. But

-7*a^2+10*a*b-7*b^2

(4)

d2:=discrim(%,a);  # should be >= 0 (otherwise d2<0 implies d1<0).  ==>

-96*b^2

(5)

b:=0;  # contradiction.

0

(6)

solve(f1,x);  # Actually:

(1/2+((1/2)*I)*7^(1/2))*a, (1/2-((1/2)*I)*7^(1/2))*a

(7)

# Contradiction. Q.E.D.

 

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