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dharr

Dr. David Harrington

8455 Reputation

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21 years, 29 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

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These are answers submitted by dharr

avoided error message but still wrong...

dharr 8455
February 25 2020
1 2

[Edited based on Carl's comment]. At least for real parameters (where the Matrix is Hermitian), there must be four eigenvectors.Not setting phi=Pi circumvents the OPs error message, and 4 eigenvectors are found, but then substituting phi=Pi leads to an error.


 

> 

restart;
> 

with(LinearAlgebra):
> 

assume(k1,real,k2,real,m1,real,m2,real,phi,real);

Leave phi unspecified for now

> 

A := <
          <0, 0, exp(I*k1) + m1, exp(I*k2) + m2>|
          <0, 0, exp(I*phi)*(exp(-I*k2) + m2), exp(-I*k1) + m1>|
          <exp(-I*k1) + m1, exp(-I*phi)*(m2 + exp(I*k2)), 0, 0>|
          <exp(-I*k2) + m2, exp(I*k1) + m1, 0, 0>
     >;

A := Matrix(4, 4, {(1, 1) = 0, (1, 2) = 0, (1, 3) = exp(-I*k1)+m1, (1, 4) = exp(-I*k2)+m2, (2, 1) = 0, (2, 2) = 0, (2, 3) = exp(-I*phi)*(exp(I*k2)+m2), (2, 4) = exp(I*k1)+m1, (3, 1) = exp(I*k1)+m1, (3, 2) = exp(I*phi)*(exp(-I*k2)+m2), (3, 3) = 0, (3, 4) = 0, (4, 1) = exp(I*k2)+m2, (4, 2) = exp(-I*k1)+m1, (4, 3) = 0, (4, 4) = 0})

Under the assumptions that the parameters are real, the Matrix is Hermitian and so must have 4 eigenvectors

> 

simplify(A-A^%H);

Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])

If we specifiy phi=Pi, we get the OP's error

> 

evals,evecs:=LinearAlgebra:-Eigenvectors(eval(A,phi=Pi)):

Error, (in LinearAlgebra:-Eigenvectors) multiplicity mismatch

If we leave phi unspecified, then we don't get the error message, and get 4 eigenvectors

> 

evals,evecs:=LinearAlgebra:-Eigenvectors(A):
nops(evals),nops(evecs);  #nops always reterns 3, as Carl points out
numelems(evals),numelems(evecs);

3, 3

4, 16

Now specify phi=Pi. Works for the eigenvalues, but not for the eigenvectors. Perhaps some manipulation could improve this, but I'm not sure

> 

evals2:=eval(evals,phi=Pi);
eval(evecs,phi=Pi);

evals2 := Vector(4, {(1) = (2+m1^2+m2^2+2*m1*cos(k1)+2*m2*cos(k2))^(1/2), (2) = -(2+m1^2+m2^2+2*m1*cos(k1)+2*m2*cos(k2))^(1/2), (3) = (2+m1^2+m2^2+2*m1*cos(k1)+2*m2*cos(k2))^(1/2), (4) = -(2+m1^2+m2^2+2*m1*cos(k1)+2*m2*cos(k2))^(1/2)})

Error, numeric exception: division by zero

 


 

Download Hermitian2.mw

need multiple curves...

dharr 8455
February 21 2020
1 0

When legend is a list there need to be multiple curves, but you had one curve with 3 points. One way to do this is:

display(pointplot([28,.6481496576],color=blue),pointplot([28,.648149657615473],color=orange),pointplot([28,.6512873548],color=red),style=point,labels = ["k", "y(k)"], legend = ["10-digit precision", "15-digit precision", "Floating-point iteration"] ,legendstyle = [font = ["HELVETICA", 9], location = right]);
 

 

invlaplace...

dharr 8455
February 19 2020
1 12

Not entirely certain I know what you want, but this may be a start.

[Edit: I changed pi (just a symbol) to Pi (the circle constant thing)]

At the beginning I define i to be sqrt(-1) as you want and D to be a regular variable rather than the differential operator.

Maple gives the answer as a sum over roots of a quartic polynomial. To get a simpler formula, you will need to give values to some variables or parhaps tell Maple their signs (using assuming)

Download Linear_System.mw

also assignment...

dharr 8455
December 07 2019
1 0

As well as Joe's corrections, you need to assign to y (y:=1). 
 

> 

r:=proc(x)
    local y;
    if (x=1)
    then    
    y:=1;
    else
        y:=0;
    end if;
    return y
    end proc;    
 

proc (x) local y; if x = 1 then y := 1 else y := 0 end if; return y end proc

> 

h:=proc(x);
    if (x= infinity)
    then    
    return 1 ;
    else
        return 0;
        end if;
    end proc

proc (x) if x = infinity then return 1 else return 0 end if end proc

> 

r(1);r(2);

1

0

> 

h(1);h(infinity);

0

1

> 

 


 

Download ww.mw

 

gcdex...

dharr 8455
December 01 2019
2 0

I suspect you want gcdex(x^9-a, b*x^6-c, x); but the answer is 1. You may need to specify the numerical value of more of the coefficients.

delayed evaluation...

dharr 8455
November 28 2019
0 2

Try

T := unapply('fsolve'(m(t, c) = -1, t = 0 .. (1/2)*Pi), c, numeric);

In general use unapply to make functions from expressions earlier in the worksheet. The numeric option means that plot and other functions evaluated with a symbolic value will not throw an error.

bc at infinity...

dharr 8455
November 25 2019
0 0

So the constant _C3 means another condition is required. Presumably you want the solution to drop off at y=infinity.

pdetest suggests your solution does not solve the pde.
 

Download pde.mw

assignment...

dharr 8455
October 27 2019
1 1

Use := to assign the right-hand side to the left. Some other issues also:

> 

v:= <1 |1| 0>;
w:=<1,1,0>;
M:=<1, 2, 3; 5, 4, 3; 7, 9, 0>;
c:=v.M;
d:=M.w;

v := Vector[row](3, {(1) = 1, (2) = 1, (3) = 0})

w := Vector(3, {(1) = 1, (2) = 1, (3) = 0})

M := Matrix(3, 3, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 3, (2, 1) = 5, (2, 2) = 4, (2, 3) = 3, (3, 1) = 7, (3, 2) = 9, (3, 3) = 0})

c := Vector[row](3, {(1) = 6, (2) = 6, (3) = 6})

d := Vector(3, {(1) = 3, (2) = 9, (3) = 16})

> 

 


 

Download vectors.mw

code for less than or equal...

dharr 8455
October 26 2019
0 0

Try entering in 1D Maple input, e.g. at a > prompt use ctrl-M then `&le;`

You will see that `&le;` is rendered as less-than-or-equals, as it would be in HTML.

missing * and extra spaces...

dharr 8455
October 18 2019
0 1

Changes in red.
 

> 

 RK2skritt:=proc(FR::procedure,xo,yo,vxo,vyo,h)
 local x, y, vx, vy, r, kx1, kx2, kx3, kx4, ky1, ky2, ky3, ky4, lx1, lx2, lx3, lx4, ly1, ly2, ly3, ly4, tmp;
 r:=sqrt(xo*xo+yo*yo);
 tmp:=-(h*FR(r))/(r);
 lx1:=h*vxo;
 ly1:=h*vyo;
 kx1:=tmp*xo;
 ky1:=tmp*yo;
 lx2:=h*(vxo+0.5*kx1);
 ly2:=h*(vyo+0.5*ky1);
 r:=sqrt((xo+0.5*lx1)^(2)+(yo+0.5*ly1)^(2));
 tmp:=-(h*FR(r))/(r);
 kx2:=tmp*(xo+0.5*lx1);
 ky2:=tmp*(yo+0.5*ly1);
 lx3:=h*(vxo+0.5*kx2);
 ly3:=h*(vyo+0.5*ky2);
 r:=sqrt((xo+0.5*lx2)^(2)+(yo+0.5*ly2)^(2));
 tmp:=-(h*FR(r))/(r);
 kx3:=tmp*(xo+0.5*lx2);
 ky3:=tmp*(yo+0.5*ly2);
 lx4:=h*(vxo+kx3);
 ly4:=h*(vyo+ky3);
 r:=sqrt((xo+lx3)^(2)+(yo+ly3)^(2));
 tmp:=-(h*FR(r))/(r);
 kx4:=tmp*(xo+lx3);
 ky4:=tmp*(yo+ly3);  #edit - one more here
 x:=xo+(lx1+2*lx2+2*lx3+lx4)/(6);
 y:=yo+(ly1+2*ly2+2*ly3+ly4)/(6);
 vx:=vxo+(kx1+2*kx2+2*kx3+kx4)/(6);
 vy:=vyo+(ky1+2*ky2+2*ky3+ky4)/(6);
 [x,y,vx,vy]; end proc:
> 

 

 

Download proc.mw

Works...

dharr 8455
October 17 2019
1 11

There is a missing *. The differentiation works in Maple 2017.
 

Download Diff.mw

 

 

a start...

dharr 8455
October 02 2019
2 1
> 

restart

Case 1

> 

de1:=diff(U(x),x,x)=0;
bc1:=U(0)=10,U(L)=20;
ans1:=dsolve({de1,bc1},U(x));

diff(diff(U(x), x), x) = 0

U(0) = 10, U(L) = 20

U(x) = 10*x/L+10

Case 2

> 

de2:=diff(U(x),x,x)=-x^2;
bc2:=U(0)=T,D(U)(L)=0;
ans2:=dsolve({de2,bc2},U(x));

diff(diff(U(x), x), x) = -x^2

U(0) = T, (D(U))(L) = 0

U(x) = -(1/12)*x^4+(1/3)*L^3*x+T

> 

 

> 

 


 

Download rod.mw

use Vectors...

dharr 8455
October 01 2019
0 2

Like this?

NULL

> 

restart
> 

n := 4; C := proc (K) options operator, arrow; Vector([seq(cos(2*Pi*K*l/n), l = 0 .. n-1)]) end proc; S := proc (K) options operator, arrow; Vector([seq(sin(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

4

proc (K) options operator, arrow; Vector([seq(cos(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

proc (K) options operator, arrow; Vector([seq(sin(2*Pi*K*l/n), l = 0 .. n-1)]) end proc

> 

C(2); S(3)

Vector[column]([[1], [-1], [1], [-1]])

Vector[column]([[0], [-1], [0], [1]])

These two are orthogonal

> 

C(2).S(3)

0

> 

C(2).C(2)

4

> 

 

NULL


 

Download orthogonal.mw

laplace and solve...

dharr 8455
September 22 2019
1 0

I did G11; the others are similar

Transfer_function.mw

Document properties...

dharr 8455
September 18 2019
2 1

In the File Menu under Document Properties, you should have an attribute named Active which is false for a regular help page and true for an active worksheet.

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