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These are answers submitted by vv

D...

vv 13143
June 30 2022
1 2 
g := (x, T) -> T*x + x^2:  #just an example
dgdx:=(x,T) -> D[1](g)(x,T);
dgdx(1, 2);
dgdx(x, 1);
g:='g';
dgdx(x, T);
convert(%,diff);


                dgdx := (x, T) -> D[1](g)(x, T)
                               4
                            1 + 2 x
                             g := g
                         D[1](g)(x, T)
                           d         
                          --- g(x, T)
                           dx        
 

asympt...

vv 13143
June 30 2022
3 2

It is obvious that the result of odetest cannot be 0, because sol is a truncated series.
(strictly speaking, it's not a `series`, the type being `+`).
But this works as expected:

restart;
Order:=20:
ode:=x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0:
sol:=dsolve(ode,y(x),'series',x=infinity):
odetest(sol,ode):
asympt(%,x); # just simplifies

            

Statistics...

vv 13143
June 29 2022
3 0

stats is obsolete; use Statistics

> 

restart;
> 

with(Statistics):
> 

f := PDF( Normal(0, 2*4.47), x );

0.3155422726e-1*2^(1/2)*exp(-0.6255974455e-2*x^2)

 (1)
> 

plot(f, x=-20..20);

 
> 

int(x*f, x);

-3.566543986*exp(-0.6255974455e-2*x^2)

 (2)
> 

int(x*f, x=0..5);

.5163728648

 (3)
> 

 

> 

 

 

Download stat.mw

The new ode...

vv 13143
June 28 2022
2 1
 
> 

 

> 

################################
# Solution "by hand"
# For simplicity, take a=0, b=1;
################################
> 

restart;
> 

de := diff(u(x),x$4) = Heaviside(x - a)*u(x) - Heaviside(x - b)*u(x);

diff(diff(diff(diff(u(x), x), x), x), x) = Heaviside(x-a)*u(x)-Heaviside(x-b)*u(x)

 (1)
> 

a:=0;b:=1;

0

  

1

 (2)
> 

dsolve(de, parametric); # wrong

u(x) = piecewise(x < 0, (1/6)*_C1*x^3+(1/2)*_C2*x^2+_C3*x+_C4, x < 1, _C1*exp(-x)+_C2*exp(x)+_C3*sin(x)+_C4*cos(x), 1 <= x, (1/6)*_C1*x^3+(1/2)*_C2*x^2+_C3*x+_C4)

 (3)
> 

de1:=eval(de) assuming x<0;
de2:=eval(de) assuming 0<x,x<1;
de3:=eval(de) assuming 1<x;

diff(diff(diff(diff(u(x), x), x), x), x) = 0

  

diff(diff(diff(diff(u(x), x), x), x), x) = u(x)

  

diff(diff(diff(diff(u(x), x), x), x), x) = 0

 (4)
> 

s1:=rhs(dsolve(de1)):
s2:=rhs(dsolve(de2)):
s3:=rhs(dsolve(de3)):
> 

S:=piecewise(x<0, eval(s1,[_C1=c1,_C2=c2,_C3=c3,_C4=c4]), x<1, eval(s2,[_C1=c5,_C2=c6,_C3=c7,_C4=c8]), eval(s3,[_C1=c9,_C2=c10,_C3=c11,_C4=c12]));

S := piecewise(x < 0, (1/6)*c1*x^3+(1/2)*c2*x^2+c3*x+c4, x < 1, c5*exp(-x)+c6*exp(x)+c7*sin(x)+c8*cos(x), (1/6)*c9*x^3+(1/2)*c10*x^2+c11*x+c12)

 (5)
> 

seq(seq(limit(diff(S,[x$k]), x=p,left) - limit(diff(S,[x$k]), x=p,right), k=0..3), p=[a,b]):
> 

solve([%]):
> 

SOL:=simplify(eval(S, %));

SOL := piecewise(x < 0, (1/6)*(-c5+c6-c7)*x^3+(1/6)*(3*c5+3*c6-3*c8)*x^2+(1/6)*(-6*c5+6*c6+6*c7)*x+c5+c6+c8, x < 1, c5*exp(-x)+c6*exp(x)+c7*sin(x)+c8*cos(x), 1 <= x, (1/6)*(-x^3*c7+(3*c7-3*c8)*x^2+(3*c7+6*c8)*x-5*c7+3*c8)*cos(1)+(1/6)*(c8*x^3+(-3*c7-3*c8)*x^2+(6*c7-3*c8)*x+3*c7+5*c8)*sin(1)-(1/6)*c5*(x^3-6*x^2+15*x-16)*exp(-1)+(1/6)*c6*exp(1)*(x^3+3*x+2))

 (6)
> 

indets(SOL, name);

{c5, c6, c7, c8, x}

 (7)
> 

plot(eval(SOL,[c5=5,c6=6,c7=7,c8=8]), x=-1..2);

 
> 

 


Download byhand-ode.mw

A way...

vv 13143
June 25 2022
1 2 
{indices}(table(AA),'pairs');

 

degree 5...

vv 13143
June 25 2022
1 0

Your equation is polynomial of degree 5 in the unknown R, with huge symbolic coefficients.
Such equations usually do not have explicit solutions (see Abel–Ruffini theorem theorem and Galois theory). 

Corrupted worksheet...

vv 13143
June 25 2022
1 1

The worksheet is corrupted, or contains some hidden fields.
If the .mw is exported as .mpl and then opened in a fresh session, it runs OK.

 

maybe...

vv 13143
June 13 2022
1 0

The documentation of SubgroupLattice says:

The algorithm used does not find any perfect subgroups of G, other than potentially G itself, its soluble residual, and the trivial group. If the soluble residual of G is of size at least 360, then it may contain other nontrivial perfect subgroups. These will not be found by the algorithm.

Maybe there are other omissions.
 

Digits...

vv 13143
June 12 2022
2 1

The denominator has many very small values ==> large roundoff errors.
Increasing Digits (e.g. Digits:=15) solves the problem.

evalf...

vv 13143
June 08 2022
2 1

If the matrix A has both exact and float entries, then it is obvious that some computations (e.g. the value of a discontinuous function) may return "unexpected" results.
Rank is such a discontinuous function. You have to accept this fact.
E.g. if A has two columns such that the first one is symbolic with at least an irrational entry,
and the second column is nonzero with float entries, the rank(A) is always 2.

> 

restart;
> 

v:=<sqrt(3), 2, 0>:
> 

A:=<v|evalf[25](v)>;

Matrix(3, 2, {(1, 1) = sqrt(3), (1, 2) = 1.732050807568877293527446, (2, 1) = 2, (2, 2) = 2., (3, 1) = 0, (3, 2) = 0.})

 (1)
> 

LinearAlgebra:-Rank(A);        # exact

2

 (2)
> 

LinearAlgebra:-Rank(evalf(A)); # "approx" but probably "expected"

1

 (3)

 

Just replace in param1,...,param4: ...

vv 13143
June 06 2022
1 0

Just replace in param1,...,param4:  alpha*gamma=0.004;   with   gamma=0.004/alpha;
(or use algsubs).

Advices:
- don't load packages which are not needed
- gamma is a Maple constant, don't use it as a variable
- linalg is deprecated; use LinearAlgebra instead.

 

Int...

vv 13143
June 06 2022
4 3 
convert(EllipticK(x), Int): 
eval(%,x=1); value(%);

       

          infinity

size...

vv 13143
June 04 2022
2 2 
with(GroupTheory):
g:=DrawSubgroupLattice(SmallGroup(200, 31), labels = ids, output=graph):
GraphTheory:-DrawGraph(g, size=[1200,800]);