## How to factorize to matrix ...

from determinant's polynomial?

## Problem with wave equation...

I am trying to find a general solution to the 1D-wave equation

Eq1:=diff(u(x,t),t\$2)=diff(u(x,t),x\$2);

pdsolve(Eq1,HINT=f(x)*g(t)); # Hinting pdsolve gives general solution using separation of variables

pdsolve({Eq1,u(x,0)=f(x),D[2](u)(x,0)=g(x)}); # without HINT and using intial conditions, I get travelling wave solution

pdsolve({Eq1,u(x,0)=f(x),D[2](u)(x,0)=g(x)},HINT=f(x)*g(t)); # Now when I try to use hint and ICs both, pdsolve return nothing.

I want to use separation of variables to find solution to the wave equation.

Any comment?

Thanks

## How to solve this differential equation ...

without _Y(t) and DESol?

f := diff(u(t), t\$2) + (2/t-1/t^2*p(1/t))*diff(u(t),t) + 1/t^4*q(1/t)*u(t) = 0;
dsolve(f,u(t));
u(t) = DESol({q(1/t)*_Y(t)/t^4+(2/t-p(1/t)/t^2)*(diff(_Y(t), t))+diff(_Y(t), t, t)}, {_Y(t)})

just would like to find u1 and u2 solution

## How to find back equation...

from galois group function's result?

## How to plot this graph?...

g[1] := (diff(a(t), t))/(t^2-1) = 1;
g[2] := (diff(a(t), t))*(diff(b(t), t)) = 1;
dsolve({eq2, eq3});
with(DynamicSystems):
sys := DiffEquation([g[1]=1, g[2]=1], inputvariable = [b(t)], outputvariable = [a(t), b(t)]):
ts := 0.1:
t_sim := 10.0:
#in_t := Sine(1, 1, 0, 0):
#in_z := Sine(1, 1, 0, 0, samplecount = round(t_sim/ts), sampletime = ts, discrete):
in_t := t:
sol := Simulate(sys, [in_t]):
p1 := plots[odeplot](sol, [[t, a(t)]], t = 0 .. t_sim, numpoints = 200, color = red):
Error, (in DEtools/convertsys) unable to convert to an explicit first-order system
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

## How to pdsolve and plot this graph?...

would like to draw the graph of x^2-1

with below sys instead of x^2-1

sys := {-(1/2)*(-x-1+sqrt(-3*x^2-2*x-3))*(diff(y(a, b), b))/x^2+(diff(y(a, b), a))*(diff(y(a, b), b)), (1/2)*(x+1+sqrt(-3*x^2-2*x-3))*(diff(y(a, b), a))/x^2+(x^2+x+1)/x^4, (1/2)*(x+1+sqrt(-3*x^2-2*x-3))*(diff(y(a, b), a))/x^2-(1/2)*(-x-1+sqrt(-3*x^2-2*x-3))*(diff(y(a, b), b))/x^2+(x^2+x+1)/x^4};
IBC:={x=1,y(a,3)=(3-1)*(3+1),y(0,b)=0,y(1,b)=0}:
pds:=pdsolve(sys,IBC,numeric,spacestep=140);
plots[display]([seq(pds:−plot(v,t=i10),i=0..5)]):

## Optimization problem...

I'm having trouble with using constrained optimization to solve a problem.

The problem is set up as:

Max: w=x1/2 y1/4

Subject to: K=x+4. K is a constant.

I'm not sure where to start.

## Is it possible to create a mapping ...

from permutation group to permutation group and inverse this mapping?

how to do?

## How to find these orbits?...

how to find [[1,3,2,4],[1,2]] and [2,3][[1,3,2,4],[1,2]] and [2,4],[[1,3,2,4],[1,2]]?

and why Orbits(G) put permutation group into power, how to display the result of this power group?

i use this or quartic polynomials' resolvent

with(GroupTheory):
G := SpecialUnitaryGroup(3, 1);
Orbits(G);
Orbit(1,G);
Orbit(3,G);

## Error when calculating orbit...

with(GroupTheory):
G := Group({[[1, 2]], [[3, 4]]});
Orbit(G);
G := Group({[[1, 2], [3, 4]]});
Orbit(G);

GroupTheory:-PermutationGroup(

{module () local cycles, p, d, work; option object; end module,

module () local cycles, p, d, work; option object; end module},

degree = 4)
Error, invalid input: GroupTheory:-Orbit expects its 1st argument, point, to be of type posint, but received module () local labels, minSupp, maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder, doDerivedSeries, doLowerCentralSeries, Intersection2, RightCosetRepresentatives, LeftCosetRepresentatives, PRA, `Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial; export generator_list, n, supergroup, Sylows, pCores, ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits, IsTransitive, Transitivity, IsPrimitive, GroupOrder, Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass, ConjugacyClasses, CayleyTable, Centre, DerivedSubgrou...
GroupTheory:-PermutationGroup(

{module () local cycles, p, d, work; option object; end module},

degree = 4)
Error, invalid input: GroupTheory:-Orbit expects its 1st argument, point, to be of type posint, but received module () local labels, minSupp, maxSupp, suppSize, AtkinsonsAlgorithm, IsSimpleGroupOrder, doDerivedSeries, doLowerCentralSeries, Intersection2, RightCosetRepresentatives, LeftCosetRepresentatives, PRA, `Giant?`, `Even?`, doStab1, doStab, CycleIndexMonomial; export generator_list, n, supergroup, Sylows, pCores, ModulePrint, ModuleDeconstruct, Generators, Orbit, Orbits, IsTransitive, Transitivity, IsPrimitive, GroupOrder, Elements, IsAbelian, IsElementary, IsSimple, ConjugacyClass, ConjugacyClasses, CayleyTable, Centre, DerivedSubgrou...

## How to calculate this resolvent?...

how to calculate Ferrari resolvent of x^4-c1*x^3+c2*x^2-c3*x+c4

## IdealMembership Testing...

I was using Maple18 for the Ideal Membership Problem. While checking it I got the following error

Error, (in F4:-GroebnerBasis) argument `[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-48,-48,48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]/A1[119295][119295]` is incorrect or out of order

Please tell me, how can I resolve this error ?.

Thank You.

## Error in Subgroup...

i use normaliser's example's code in maple help file

generators is [50] originally, then i calculated again , it become [51], [52], [53] , i do not know whether virus change my library

then i use another computer to calculate, the result is [50]

then i further calculate subgroup got error below

with(GroupTheory):
with(group):
G := AlternatingGroup(5);
IsFinite(G);
GroupOrder(G);
spg := SylowSubgroup(5, G);
IsAbelian(spg);
Elements(spg);
lprint(%);
H := Subgroup(Elements(G), spg);
N := Normaliser(G, spg);
#N := Normaliser(spg, G);
Elements(N);
lprint(%);
Elements(G);
H2 := Subgroup({[[5,2],[3,4]]}, G);
H2 := Subgroup(Elements(G), G);
elements2 := convert(Elements(G), 'list');
generators := map(ListTools:-Search, [Perm([[1,2,3]])], elements2);
H2 := Subgroup(generators, G);

H2 := Subgroup(Perm([generators]), G);
Error, invalid input: GroupTheory:-Subgroup expects its 1st argument, generators, to be of type {list, set, identical(undefined)}, but received module () local cycles, p, d, work; option object; end module
H2 := Subgroup(generators, G);
Error, (in Perm:-normalform) invalid input: map expects 2 or more arguments, but received 1

SubgroupMembership(H2, G);

## Eigenvalues of a huge matrix...

I faced a very large eigenproblem during my research. The square matrix under consideration is of size more than 2^30 times 2^30. I have tried to deal with this problem by the QR algorithm with double implicit shift (more precisely, the Francis double step QR algorithm). I'm a very beginner of programming, but I tried as follows:

--------------------------------------------------------------------------------------------------

A := Matrix([[7, 3, 4, -11, -9, -2], [-6, 4, -5, 7, 1, 12], [-1, -9, 2, 2, 9, 1], [-8, 0, -1, 5, 0, 8], [-4, 3, -5, 7, 2, 10], [6, 1, 4, -11, -7, -1]]):
H := HessenbergForm(A):
p:=6:
for p while p>2 do:
q:=p-1:
s:=H(q,q)+H(p,p):
t:=H(q,q)*H(p,p)-H(q,p)*H(p,q):
x:=(H(1,1))^(2)+H(1,2)*H(2,1)-s*H(1,1)+t:
y:=H(2,1)*(H(1,1)+H(2,2)-s):
z:=H(2,1)*H(3,2):
for k from 0 to p-3 do:
V:=Vector([x,y,z]):
P:=Transpose(HouseholderMatrix(1/(Norm(V+exp(argument(V(1))*I)*Norm(V,2)*Vector(3,shape=unit[1]),2))*(V+exp(argument(V(1))*I)*Norm(V,2)*Vector(3,shape=unit[1])))):
r:=max(1,k):
H[k+1..k+3,r..6]:=MatrixMatrixMultiply(Transpose(P),SubMatrix(H,[k+1..k+3],[r..6])):
r:=min(k+4,6):
H[1..r,k+1..k+3]:=MatrixMatrixMultiply(SubMatrix(H,[1..r],[k+1..k+3]),P):
x:=H(k+2,k+1):
y:=H(k+3,k+1):
if k<3 then z:=H(k+4,k+1):
end if:
od:
P:=GivensRotationMatrix(Vector([x,y]),1,2):
H[q..p,p-2..6]:=MatrixMatrixMultiply(Transpose(P),SubMatrix(H,[q..p],[p-2..6])):
H[1..p,p-1,p]:=MatrixMatrixMultiply(SubMatrix(H,[1..p],[p-1,p]),P):
if abs(H(p,q))<10^(-20)*(abs(H(q,q))+abs(H(p,p))) then    H(p,q):=0: p:=p-1:q=p-1:
elif abs(H(p-1,q-1))<10^(-20)*(abs(H(q-1,q-1))+abs(H(q,q))) then    H(p-1,q-1):=0: p:=p-2:q:=p-1:
end if:  od:
--------------------------------------------------------------------------------------------------

It seemed that replacing 0 in a Hessenberg matrix by a non-zero element is not allowed. How can I remedy this?

Plus, can anyone tell me the problem of the above thing(it's not really a programming...;( ), please?

I would also appreciate it if someone let me know a better idea for a huge eigenproblem.