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Alec Mihailovs

Dr. Aleksandrs Mihailovs

4495 Reputation

21 Badges

20 years, 337 days
Mihailovs, Inc.
Owner, President, and CEO
Tyngsboro, Massachusetts, United States

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Contact Dr. Aleksandrs Mihailovs

I received my Ph.D. from the University of Pennsylvania in 1998 and I have been teaching since then at SUNY Oneonta for 1 year, at Shepherd University for 5 years, at Tennessee Tech for 2 years, at Lane College for 1 year, and this year I taught at the University of Massachusetts Lowell. My research interests include Representation Theory and Combinatorics.

MaplePrimes Activity


These are replies submitted by Alec Mihailovs

You are welcome!...

Alec Mihailovs 4495
October 18 2006
You are welcome! I like to read replies like that - and I should say that they are very rare. In most cases I don't get any reply at all (and sometimes replies are quite insulting.)

Quite simple...

Alec Mihailovs 4495
October 18 2006
0 0
Actually, it can be done quite simple, 
r:=rand(8):
G:=GF(2,3,x^3+x+1):
M:=Matrix(3,()->G[ConvertIn](x^r()));

                     [       2           2         2]
                     [  x + x       1 + x     1 + x ]
                     [                              ]
                M := [                   2          ]
                     [    x         x + x       1   ]
                     [                              ]
                     [         2                 2  ]
                     [1 + x + x       x         x   ]
                
T:=a->map(c->modp1(ConvertOut(c,x),2),a):
`&*`:=(a,b)->map(G[ConvertIn],T(a).T(b)):
DetG:=G[ConvertIn]@LinearAlgebra:-Determinant@T:
Ad:=a->map(G[ConvertIn],LinearAlgebra:-Adjoint(T(a))):
Inv:=M->map(G[`*`],Ad(M),G[inverse](DetG(M))):
For example, 
Inv(M);

                   [                   2          ]
                   [    x         1 + x       1   ]
                   [                              ]
                   [         2         2          ]
                   [1 + x + x     1 + x       x   ]
                   [                              ]
                   [                             2]
                   [    0         1 + x     1 + x ]

Inv(M) &* M;

                            [1    0    0]
                            [           ]
                            [0    1    0]
                            [           ]
                            [0    0    1]

DetG(M);
                                     2
                                x + x

Quite simple...

Alec Mihailovs 4495
October 18 2006
Actually, it can be done quite simple, 
r:=rand(8):
G:=GF(2,3,x^3+x+1):
M:=Matrix(3,()->G[ConvertIn](x^r()));

                     [       2           2         2]
                     [  x + x       1 + x     1 + x ]
                     [                              ]
                M := [                   2          ]
                     [    x         x + x       1   ]
                     [                              ]
                     [         2                 2  ]
                     [1 + x + x       x         x   ]
                
T:=a->map(c->modp1(ConvertOut(c,x),2),a):
`&*`:=(a,b)->map(G[ConvertIn],T(a).T(b)):
DetG:=G[ConvertIn]@LinearAlgebra:-Determinant@T:
Ad:=a->map(G[ConvertIn],LinearAlgebra:-Adjoint(T(a))):
Inv:=M->map(G[`*`],Ad(M),G[inverse](DetG(M))):
For example, 
Inv(M);

                   [                   2          ]
                   [    x         1 + x       1   ]
                   [                              ]
                   [         2         2          ]
                   [1 + x + x     1 + x       x   ]
                   [                              ]
                   [                             2]
                   [    0         1 + x     1 + x ]

Inv(M) &* M;

                            [1    0    0]
                            [           ]
                            [0    1    0]
                            [           ]
                            [0    0    1]

DetG(M);
                                     2
                                x + x

map...

Alec Mihailovs 4495
October 17 2006
Also, it is convenient to use map for producing a list from a list, 
plotList:=map(plot,fns,x=-3..3):
By the way, in the described situation the plot can be created without display, just as 
plot(fns,x=-3..3);
that has an additional advantage of using different colors for different curves.

{ from the lhs...

Alec Mihailovs 4495
October 17 2006
0 0
It is possible to add { from the lhs, 
s:=MathML:-Export(piecewise(x+2*y=0,``,x+4*y+3*z=5,``,x+y+z=0,``)):
MathML:-Import(s);

              PIECEWISE([``, x+2*y = 0],[``, x+4*y+3*z = 5],[``, x+y+z = 0])

{ from the lhs...

Alec Mihailovs 4495
October 17 2006
It is possible to add { from the lhs, 
s:=MathML:-Export(piecewise(x+2*y=0,``,x+4*y+3*z=5,``,x+y+z=0,``)):
MathML:-Import(s);

              PIECEWISE([``, x+2*y = 0],[``, x+4*y+3*z = 5],[``, x+y+z = 0])

axis[2]...

Alec Mihailovs 4495
October 17 2006
0 0
In Standard Maple, you can set its color to white, like in the following example, 
plot(x^2-1,x=-2..2,axis[2]=[color=white]);

axis[2]...

Alec Mihailovs 4495
October 17 2006
In Standard Maple, you can set its color to white, like in the following example, 
plot(x^2-1,x=-2..2,axis[2]=[color=white]);

LaTeX...

Alec Mihailovs 4495
October 17 2006
0 0
Manuals were produced by exporting Maple (command line or Classic Maple) to LaTeX, not by using the Document mode.

LaTeX...

Alec Mihailovs 4495
October 17 2006
Manuals were produced by exporting Maple (command line or Classic Maple) to LaTeX, not by using the Document mode.

With colors or with a grid...

Alec Mihailovs 4495
October 17 2006
0 0
It can be done with colors as 
with(plots):
a:=implicitplot([2*R^3-R^2+5*R^2*q+4*R*q^2+q^3=0,
                 q^3+3*R*q^2-R^2+3*R^2*q+R^3=0,
                 R^2+2*R^3+3*R^2*q-q^3=0,
                 q^2+R=.3],R=0..2,q=0..1,
   color=[red,green,blue,black],numpoints=100000):
f:=piecewise(2*R^3-R^2+5*R^2*q+4*R*q^2+q^3>0 and
   q^3+3*R*q^2-R^2+3*R^2*q+R^3>0 and R^2+2*R^3+3*R^2*q-q^3>0 and
   q^2+R>.3,1,0):
b:=implicitplot(f>0.5,R=0..2,q=0..1,filled=true,
                numpoints=100000,coloring=[green,white]):
display(a,b);
What you wanted exactly, can be done as 
with(plots):
a:=implicitplot([2*R^3-R^2+5*R^2*q+4*R*q^2+q^3=0,
                 q^3+3*R*q^2-R^2+3*R^2*q+R^3=0,
                 R^2+2*R^3+3*R^2*q-q^3=0,
                 q^2+R=.3],R=0..2,q=0..1,
   color=[red,green,blue,black],numpoints=100000):
with(plottools):
c:=seq(line([max(.3-q^2,0),q],[2,q]),q=0..1,0.05):
d:=seq(line([R,max(fsolve(q^3+3*R*q^2-R^2+3*R^2*q+R^3,q),0)],
      [R,min(fsolve(R^2+2*R^3+3*R^2*q-q^3,q),1)]),R=0..2,0.1):
display(a,c,d);

With colors or with a grid...

Alec Mihailovs 4495
October 17 2006
It can be done with colors as 
with(plots):
a:=implicitplot([2*R^3-R^2+5*R^2*q+4*R*q^2+q^3=0,
                 q^3+3*R*q^2-R^2+3*R^2*q+R^3=0,
                 R^2+2*R^3+3*R^2*q-q^3=0,
                 q^2+R=.3],R=0..2,q=0..1,
   color=[red,green,blue,black],numpoints=100000):
f:=piecewise(2*R^3-R^2+5*R^2*q+4*R*q^2+q^3>0 and
   q^3+3*R*q^2-R^2+3*R^2*q+R^3>0 and R^2+2*R^3+3*R^2*q-q^3>0 and
   q^2+R>.3,1,0):
b:=implicitplot(f>0.5,R=0..2,q=0..1,filled=true,
                numpoints=100000,coloring=[green,white]):
display(a,b);
What you wanted exactly, can be done as 
with(plots):
a:=implicitplot([2*R^3-R^2+5*R^2*q+4*R*q^2+q^3=0,
                 q^3+3*R*q^2-R^2+3*R^2*q+R^3=0,
                 R^2+2*R^3+3*R^2*q-q^3=0,
                 q^2+R=.3],R=0..2,q=0..1,
   color=[red,green,blue,black],numpoints=100000):
with(plottools):
c:=seq(line([max(.3-q^2,0),q],[2,q]),q=0..1,0.05):
d:=seq(line([R,max(fsolve(q^3+3*R*q^2-R^2+3*R^2*q+R^3,q),0)],
      [R,min(fsolve(R^2+2*R^3+3*R^2*q-q^3,q),1)]),R=0..2,0.1):
display(a,c,d);

Implicitplot and coloring...

Alec Mihailovs 4495
October 17 2006
A region given by an inequality can be colored using implicitplot command. For example, 
plots[implicitplot](x^2+y^2<1,x=-2..2,y=-2..2,filled=true);
Colors can be changed by using coloring option. For example, 
plots[implicitplot]((x-0.5)^2+(y-0.5)^2<1,x=-2..2,y=-2..2,filled=true,
coloring=[green,white]);
For several inequalities it can be done as in the following example, 
with(plots):
a:=implicitplot([2*R^3-R^2+5*R^2*q+4*R*q^2+q^3=0,
                 q^3+3*R*q^2-R^2+3*R^2*q+R^3=0,
                 R^2+2*R^3+3*R^2*q-q^3=0,
                 q^2+R=.3],R=0..2,q=0..1,
   color=[red,green,blue,black],numpoints=100000):
f:=piecewise(2*R^3-R^2+5*R^2*q+4*R*q^2+q^3>0 and
   q^3+3*R*q^2-R^2+3*R^2*q+R^3>0 and R^2+2*R^3+3*R^2*q-q^3>0 and
   q^2+R>.3,1,0):
b:=implicitplot(f>0.5,R=0..2,q=0..1,filled=true,
                numpoints=100000,coloring=[green,white]):
display(a,b);

That's not that easy...

Alec Mihailovs 4495
October 17 2006
0 0
I don't see how that can be easily done. That, perhaps, goes into 'Product Suggestions' category. From other point of view, in this particular example, I would click Text at the beginning of the line, type 
This is the formula that we have been looking for,
then switch to Math, enter the formula and click enter button. Then the formula would appear in the middle of the next line and would be properly labeled. To avoid duplication, the expression at the end of the text can be deleted. It is not what you wanted, but in general, it is a better style.

That's not that easy...

Alec Mihailovs 4495
October 17 2006
I don't see how that can be easily done. That, perhaps, goes into 'Product Suggestions' category. From other point of view, in this particular example, I would click Text at the beginning of the line, type 
This is the formula that we have been looking for,
then switch to Math, enter the formula and click enter button. Then the formula would appear in the middle of the next line and would be properly labeled. To avoid duplication, the expression at the end of the text can be deleted. It is not what you wanted, but in general, it is a better style.
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