We can easily get the result as  piecewise  function:

Download int.mw

If you want an undisclosed sum, then use  Sum  instead of  sum :

E:=(2*h^2*Sum(x[i]^2,i=1..4))*alpha+(2*h^2*Sum(x[i]^2*y[i],i=1..4))*beta+2*h*Sum(x[i]^2,i=1..4)-2*h*Sum(x[i]*x[i+1],i=1..4);
map(t->2*t,collect(E/2, h));

The visualization shows that Maple returns the correct result, which can be simplified (see the last line of the code):

Download positive_function_new2.mw

This can be done in many ways. Here are 2 ways:

Download heightofcylinder-new.mw

Your matrix equation  S*A*inverse(S)=A  is equivalent to  S*A-A*S=0  provided that  det(S)<>0 .
The solution is below:

Download ME.mw

Let  k=theta/V  so  theta=k*V .

isolate(subs(theta = k*V, (1/3)*m*a^2*theta*s^2+(1/16)*m*g*a*sqrt(2)*(4+4*theta-Pi)+theta*B*s = (-K__m^2*s*theta+K__m*V)/(L*s+R)), k);
 ;

In real domain use the  surd  function for this:

The reason that the blue curve is not completely plotted in your document  is that Maple, for a negative number of the three possible values of the cubic root, returns the principal value of the cubic root (with the smallest argument). Compare:
 

simplify((-8)^(1/3)); 
surd(-8, 3)

Download inverseExample_new.mw

In fact, all your functions depend only on the variable  t . I made all the necessary simplifications and changes. In particular, I replaced  combine  with  simplify  with the option  size , because it turns out a more compact expression:

Download equations_new.mw

You must specify a sequence step, for example:

f := unapply(3*x^2-2*x^3-1.080674649*x^2*(x-1)^2-.8118769171*x^2*(x-1)^3+.4147046974*x^2*(x-1)^4+.4585681954*x^2*(x-1)^5, x):
ma := seq(f(x), x = 0..1, 0.1);

             ma := 0., 0.02517819625, 0.09374594496, 0.1954298021, 0.3207056009, 0.4607261850, 0.6065902371, 0.7481833253, 0.8728222816, 0.9639340323, 1.000

restart;	
f:=(x,y)->sin(x)*cos(y):
g:=(x,y)->sin(y)*cos(x):
v:=(x,y)->combine(f(x,y)-g(x,y)):
v(x,y);

alpha:=1: dT:=Th-Tc: n:=1:
plot([[Th-273,subs(Tc=400,n*alpha*dT/2),Th=300..700],[Th-273,subs(Tc=500,n*alpha*dT/2),Th=300..700]]);

Eq1.mw

restart;
V:=Vector[row]([x,y,z]):
M:=Matrix([seq(seq([`if`(y<>0,eval(solve(x*'y'+'z'=0,x),['y'=y,'z'=z]),undefined),y,z], y=-1..5),z=0..5)]):
interface(rtablesize=50):
<V; M>;

PS. "undefined" means that if  y=0  then there is no unique value  x  satisfying this equation. However, this case splits into 2 subcase: if  z = 0 , then  x  can be any number, and if  z<>0 , then this  x  does not exist. Below is a version of the code that takes these options into account:

restart;
V:=Vector[row]([x,y,z]):
M:=Matrix([seq(seq([`if`(y<>0,eval(solve(x*'y'+'z'=0,x),['y'=y,'z'=z]),`if`(z<>0,`not exist`,any)),y,z], y=-1..5),z=0..5)]):
interface(rtablesize=50):
<V; M>;

Use the tgonometric form of complex numbers:

z1 := cos(alpha)+I*sin(alpha): 
z2 := cos(beta)+I*sin(beta):
is((z1^2+2*z1*z2+z2^2)/(z1+z2) = z1/z2+z2/z1);
is(z1/z2+z2/z1+2 = 2*(1+cos(alpha-beta)));

                                          false
                                           true

I simplified both of your procedures by making the following changes:
1. The parameter of the procedures is taken as a whole vector, and not its coordinates.
2. Used elementwise operators.
3. Used two-argument  arctan .

Now everything works as expected:
 

Download PP.mw

According to Maple's help, in Maple there is no  Vector[col] , but there is  Vector[column]  (or Vector).
In Maple 2018.2, Vector[col] does not kill the session, but just an error message appears.