#hw3.txt
#Sunday, Feb 2 20014
#Ha Luu

2)read "C:/Users/Ha/Desktop/Experimental Math/C3.txt";
Digits:=10: evalf(det(Hil(50))),det(evalf(Hil(50)));
                          -1466                 -455
            1.392615569 10     , -5.096100083 10    
Digits:=20: evalf(det(Hil(50))),det(evalf(Hil(50)));
                          -1466                          -784
  1.3926155689351399681 10     , 3.9671871974393549731 10    
Digits:=30: evalf(det(Hil(50))),det(evalf(Hil(50)));
                                             -1466  
           1.39261556893513996812723740903 10     , 

                                               -1038
             5.67192762637403569656679171521 10     
    
Digits:=70: evalf(det(Hil(50))),det(evalf(Hil(50)));
1.39261556893513996812723740903442695607373829370969385193902295\

            -1466                                                
  7826920 10     , -2.304665266774485556923318788680028690505108\

                                -1463
  483311064493728155655640208 10     
Digits:=80: evalf(det(Hil(50))),det(evalf(Hil(50)));
1.39261556893513996812723740903442695607373829370969385193902295\

                      -1466                                      
  78269197711269990 10     , 1.392615431993038208497180946103077\

                                                   -1466
  0904407143969922847032457369046085405359836602 10     

###### You need at least 80 digits in the floating point so that there is a good agreement between the 2 evalf function.



3) ###Count number of inversion.
inv1:=proc(p) local i,j,n,k:
k:=0:
n:=nops(p):

for i from 1 to n do
 for j from i+1 to n do

    if p[i] > p[j] then
        k:= k+1:
    fi:

 od:
od:

k:

end:

###generating function

gf:=proc(n,q) local i,j,X,f,p:
f:=0:

X:=permute(n):


for i from 1 to n! do
	p:=inv1(X[i]):
	f:=f+q^p:
od:

f:
end:

#####    
factor(gf(1,q));
                               1
factor(gf(2,q));
                             1 + q
factor(gf(3,q));
                              / 2        \
                      (1 + q) \q  + q + 1/
factor(gf(4,q));
                 / 2    \ / 2        \        2
                 \q  + 1/ \q  + q + 1/ (1 + q) 

factor(gf(5,q));
     / 2        \ / 4    3    2        \ / 2    \        2
     \q  + q + 1/ \q  + q  + q  + q + 1/ \q  + 1/ (1 + q) 


factor(gf(6,q));
                                                           2 
  / 2        \ / 4    3    2        \ / 2    \ / 2        \  
  \q  - q + 1/ \q  + q  + q  + q + 1/ \q  + 1/ \q  + q + 1/  

           3
    (1 + q) 

factor(gf(7,q));
/ 2        \ / 4    3    2        \ / 6    5    4    3    2    
\q  - q + 1/ \q  + q  + q  + q + 1/ \q  + q  + q  + q  + q  + q

                             2         
      \ / 2    \ / 2        \         3
   + 1/ \q  + 1/ \q  + q + 1/  (1 + q) 

factor(gf(8,q));
/ 2        \ / 4    3    2        \ / 4    \ / 6    5    4    3
\q  - q + 1/ \q  + q  + q  + q + 1/ \q  + 1/ \q  + q  + q  + q 

                         2             2         
      2        \ / 2    \  / 2        \         4
   + q  + q + 1/ \q  + 1/  \q  + q + 1/  (1 + q) 


#### For each component, their exponent increase after a specific number of increasing in n :
For example, everytime n increases by 2, exponent of (q+1) increases by 1
Everytime n increases by 3, exponent of (q^2+q+1) increases by one.
And so on, however, I have difficulty predicting which completely new elements will be introduce to the equation as n in creases.


















1) Maple Mannual 30-60
f:='f':
y:=f(x);
                              f(x)
dy:=diff(y,x);
                             d      
                            --- f(x)
                             dx     
ddy:=diff(y,x,x);
                          d  / d      \
                         --- |--- f(x)|
                          dx \ dx     /
dsolve(ddy+5*dy+6*y = sin(x)*exp(-3*x),y);
              f(x) = exp(-3 x) _C2 + exp(-2 x) _C1

                   1                            
                 - - (sin(x) - cos(x)) exp(-3 x)
                   2                            
g:='g': y:=f(x): z:=g(x):
de1:= diff(y,x)+diff(z,x) =exp(x);
                / d      \   / d      \         
                |--- f(x)| + |--- g(x)| = exp(x)
                \ dx     /   \ dx     /         
de2:= diff(y,x)-3*z =x;
                    / d      \             
                    |--- f(x)| - 3 g(x) = x
                    \ dx     /             
dsolve({de1,de2,f(0)=8/9,g(0)=10/9},{y,z});
          /       1     3          3             8  
         { f(x) = - x + - exp(x) - - exp(-3 x) + -, 
          \       3     4          4             9  

                    1     1   1          3          \ 
           g(x) = - - x + - + - exp(x) + - exp(-3 x) }
                    3     9   4          4          / 
z:='z':
asympt(Psi(z),z,3);
                           1      1      /1 \
                  ln(z) - --- - ----- + O|--|
                          2 z       2    | 4|
                                12 z     \z /
plot(sin(x),x=-2*Pi..2*Pi);

plot(sec(x),x=-Pi..2*Pi);

plot(sec(x),x=-Pi..2*Pi,y=-5..5);
Error, (in plot) unexpected option: f(x) = -5 .. 5
?plot
# note: in the new maple, range restriction does not work
p1:= plot([sqrt(x),3*log(x)],x=0..400,title =`The Square Root and log functions`);

#note: display is important function to show the graph.
p2:=textplot([[360,16,`y=3log(x)`],[130,10,`y=sqrt(x)`]]):
display(p1,p2);

plotsestup(ps,plotoutput=`plot.ps`,plotoptions=`potrait,noborder`);
             plotsestup(ps, plotoutput = plot.ps, 

               plotoptions = potrait,noborder)
plot(sin(x),x=-2*Pi..2*Pi);
nterface(plotdevice=inline);
                            "inline"
plot3d(exp(-(x^2+y^2-1)^2), x=-2..2, y=-2..2);
Error, (in plot3d) bad range arguments: x = -2 .. 2, f(x) = -2 .. 2

plot3d(sin(x+y),x=-1..1, y=-1..1);

Error, (in plot3d) bad range arguments: x = -1 .. 1, f(x) = -1 .. 1


# note: could not figure out what's wrong with the plot3d eventhough following ?plot3d example


plot3d([sqrt(1+u^2)*cos(t),sqrt(1+u^2)*sin(t),u],u=-1..1,t=0..2*Pi);
#note: do not have the helix graph?


with(plots):
contourplot(exp(-(x^2+y^2-1)^2),x=-(1.3)..(1.3),y=-(1.3)..(1.3),filled=true, coloring=[blue,red]);
Error, (in plot/iplot2d/expression) bad range arguments x = -1.3 .. 1.3, f(x) = -1.3 .. 1.3


#note: keep having trouble with the bad range argument?

with(plots):
implicitplot3d(y^2-x^2 = z,x=-2..2,y=-2..2,z=-4..4);
Error, (in plots:-implicitplot3d) invalid input: `plots/iplot3d` expects its 3rd argument, r2, to be of type {range, name = range}, but received f(x) = -2 .. 2
with(plots):
animate3d([r*cos(t+a),r*sin(t+a),r^2*cos(2*t)],r=0..1,t=0..2*Pi, a=0..3, frames=10,style=patch,title=`The Rotating Saddle`);

v:=vector([1,2,3]);

A:=matrix(2,3,[a,b,c,d,e,f]);

v;
                               v
A;
                               A
print(v);
                               v
print(A);
                               A
op(A);

eval(A);

B:=matrix(2,2);

entermatrix(B);

#note: with new verson of Maple, it's pop-up window, so no semicolon needed after entering the number.
f:=(i,j)->x^(i*j);
           (i j)
(i, j) -> x     
A:=matrix(4,4,f);

factor(det(A));
                20 / 2        \        2        6
               x   \x  + x + 1/ (x + 1)  (x - 1) 
A:=matrix(2,2,[1,2,3,4]):
B:=matrix(2,2,[-2,3,-5,1]);

A+B;
                             A + B
evalm(%);

A&*B;
                             A &* B
evalm(%);