The first thing you should know is that assigning a value to a variable removes any previous assumptions on that variable. 

I'm not sure why about(P) doesn't work here.  But Maple really does know that P < 0.  For example:

> restart;
   assume(g > 0);
   P:= -g;
   signum(P);

                       -1

> rationalize( -1/((z+1+I)*(-z-1+I)));

1/(z^2+2*z+2)

> rationalize( -1/((z+1+I)*(-z-1+I)));

1/(z^2+2*z+2)

In this case, one solution is _Z = 0.  However, that doesn't work for your system of equations, because you'd have d -> infinity.  There don't seem to be any other real solutions.

In this case, one solution is _Z = 0.  However, that doesn't work for your system of equations, because you'd have d -> infinity.  There don't seem to be any other real solutions.

Velocity (as the term is usually used in physics) is a vector quantity.  The magnitude of the velocity is the speed. 

Velocity (as the term is usually used in physics) is a vector quantity.  The magnitude of the velocity is the speed. 

There is no way that collect would produce a result containing lists unless lists were present in the input.  So you should take a closer look at your L_a.

There is no way that collect would produce a result containing lists unless lists were present in the input.  So you should take a closer look at your L_a.

Actually it seems to be wrong except for _Z2 = -1 or 1.  Branch cut problems.

> eq:= -ln(1-x)/x-x+2 = 0;
   shouldbe := eval(eq, x = solve(eq, x, AllSolutions));
   [seq(evalf[20](traperror(eval(shouldbe,_Z2=j))), j=-10 .. 10)];

[5.2049707240995594251-3.7706064121630124956*I = 0., 
-2.6326734703584719931+3.2723319236118533566*I = 0., 
4.8099687621160570724-3.3314463279425902525*I = 0., 
-2.1700228302109924482+2.7751195944259940559*I = 0., 
4.3691885969554688280-2.8259690272781035975*I = 0., 
-1.6290579922541131780+2.1807825380228797609*I = 0., 
3.8635721553217173198-2.2109953536872775992*I = 0., 
-.9522458026752997077+1.3999135424388421580*I = 0., 
3.2687987773813298909-1.3612997717890983965*I = 0., 
.1e-18+.9e-19*I = 0., 
"numeric exception: division by zero", 
.1e-18-.9e-19*I = 0., 
3.2687987773813298909+1.3612997717890983965*I = 0., 
-.9522458026752997077-1.3999135424388421580*I = 0., 
3.8635721553217173198+2.2109953536872775992*I = 0., 
-1.6290579922541131780-2.1807825380228797609*I = 0., 
4.3691885969554688280+2.8259690272781035975*I = 0., 
-2.1700228302109924482-2.7751195944259940559*I = 0., 
4.8099687621160570724+3.3314463279425902525*I = 0., 
-2.6326734703584719931-3.2723319236118533566*I = 0., 
5.2049707240995594251+3.7706064121630124956*I = 0.]

 The situation seems to be this.  Multiply eq by x, and take the exponential of both sides, you get

exp(-x^2+2*x)/(1-x) = 1

What Maple returned was the general solution of this equation.  However, when you go back to the original, for _Z2 = 0 you get a division by 0 error, while for _Z2 >= 2 or <= -2 you would need to use a non-principal branch of the logarithm to make it work.

It works for me.  For example:

> resizeit:= proc()
   local s,f;
   s:= Maplets:-Tools:-Get('TF2'::posint,corrections);
   f:= Maplets:-Tools:-Get('TF1'::algebraic,corrections);
   Maplets:-Tools:-Set(PL1('height')=s);
   Maplets:-Tools:-Set('PL1'=plot(f,x=0..10));
  end proc;

  maplet := Maplet([
    ["Enter a function of 'x':", TextField['TF1']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    ["New size ",TextField['TF2']("200")],
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF1, x = 0..10)') ),
       Button("Resize",Evaluate('function'="resizeit")),
       Button("OK", Shutdown(['TF2']))
    ]
   ]):

  Maplets[Display](maplet);

It works for me.  For example:

> resizeit:= proc()
   local s,f;
   s:= Maplets:-Tools:-Get('TF2'::posint,corrections);
   f:= Maplets:-Tools:-Get('TF1'::algebraic,corrections);
   Maplets:-Tools:-Set(PL1('height')=s);
   Maplets:-Tools:-Set('PL1'=plot(f,x=0..10));
  end proc;

  maplet := Maplet([
    ["Enter a function of 'x':", TextField['TF1']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    ["New size ",TextField['TF2']("200")],
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF1, x = 0..10)') ),
       Button("Resize",Evaluate('function'="resizeit")),
       Button("OK", Shutdown(['TF2']))
    ]
   ]):

  Maplets[Display](maplet);

I think you just have to replot after resizing.  For example:

> with(Maplets[Elements]):
 maplet := Maplet([
    ["Enter a function of 'x':", TextField['TF2d']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF2d, x = 0..10)') ),
       Button("Shrink",Action(SetOption(PL1('height')=100),
                              Evaluate('PL1'='plot(TF2d, x = 0..10)'))),
       Button("OK", Shutdown(['TF2d']))
    ]
 ]):
 result := Maplets[Display](maplet);

However, if you want to expand the plot, you'll probably also have to expand the window it's in.  For example:

> maplet := Maplet(Window[W]([
    ["Enter a function of 'x':", TextField['TF2d']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF2d, x = 0..10)') ),
       Button("Expand",Action(SetOption(W('height')=550),
                              SetOption(PL1('height')=400),
                              Evaluate('PL1'='plot(TF2d, x = 0..10)'))),
       Button("OK", Shutdown(['TF2d']))
    ]
 ])):
 result := Maplets[Display](maplet);

I think you just have to replot after resizing.  For example:

> with(Maplets[Elements]):
 maplet := Maplet([
    ["Enter a function of 'x':", TextField['TF2d']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF2d, x = 0..10)') ),
       Button("Shrink",Action(SetOption(PL1('height')=100),
                              Evaluate('PL1'='plot(TF2d, x = 0..10)'))),
       Button("OK", Shutdown(['TF2d']))
    ]
 ]):
 result := Maplets[Display](maplet);

However, if you want to expand the plot, you'll probably also have to expand the window it's in.  For example:

> maplet := Maplet(Window[W]([
    ["Enter a function of 'x':", TextField['TF2d']("sin(x)")],
    Plotter['PL1']( plot(sin(x), x = 0..10),height=200,width=200 ),
    [
       Button("Plot", Evaluate('PL1' = 'plot(TF2d, x = 0..10)') ),
       Button("Expand",Action(SetOption(W('height')=550),
                              SetOption(PL1('height')=400),
                              Evaluate('PL1'='plot(TF2d, x = 0..10)'))),
       Button("OK", Shutdown(['TF2d']))
    ]
 ])):
 result := Maplets[Display](maplet);

Are you saying f(e) is quadratic in w?  So let f(e) =   A*w^2 + B*w + C.

> e := ((w-b)/b)^lambda;
  prof:= f(e) - w*L;
  prof1:= eval(prof, f(e) = A*w^2 + B*w + C);

Now you say the derivative of this wrt w is 0 at w = -b/(lambda - 1).

> eval(diff(prof1, w), w = -b/(lambda-1));

That being 0 gives you a relation between A and B, which you can solve for either of these, let's say for B.

> B0:= solve(%, B);

B0 := (2*A*b+L*lambda-L)/(lambda-1)

The maple tag is acting up again: that should be

    B0 := (2*A*b+L*lambda-L)/(lambda-1)

So now f(e) = A*w^2 + B0 * w + C.  Now.we can express w in terms of e.  But since I defined e as an expression in w, I'll use E instead.

> wE:= solve(e=E, w);

    wE := exp(ln(E)/lambda)*b+b

And then the function f is

> f(E) =  simplify(A*wE^2 + B0*wE + C);

f(E) = (A*b^2*E^(2/lambda)*lambda-A*b^2*E^(2/lambda)+2*A*b^2*E^(1/lambda)*lambda+A*b^2*lambda+A*b^2+b*L*lambda*E^(1/lambda)+b*L*lambda-b*L*E^(1/lambda)-b*L+C*lambda-C)/(lambda-1)

Is that what you had in mind?