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dharr

Dr. David Harrington

8205 Reputation

22 Badges

20 years, 335 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.

MaplePrimes Activity


These are answers submitted by dharr

sprintf...

dharr 8205
October 09 2017
1 1

sprintf("%.4f",0.0141); gives "0.0141"

The format parameters give finer control about width, precision etc see ?sprintf

A 2-D math entering mishap...

dharr 8205
September 13 2017
1 0

If you right-click on the first definition of f, and use 2-D math/convert/convert to 1-D math input you will see:

f:=k->;sum(k*x(n), n = 2 .. 3);f(1);

There is an unwanted something (appears as a semicolon here, but that's prob not what you entered) after the arrow and before the sum, i.e., this line is corrupted somehow.

parametric plot...

dharr 8205
September 09 2017
1 0

A parametric plot is an alternative to Kitonum's implicit plot, and is general for switching axes:

plot([-4.918231636*z^3+2.486861786*z^2+.8573483099*z+2.341623514,z,z=-0.5..0.8]);

and use the view option to define the plot area.

difftorec...

dharr 8205
August 24 2017
2 0

Assuming that your x should be t, then

> 

restart;
> 

EQ:=(3*t)*diff(y(t),t,t)+y(t)=0;
gfun:-diffeqtorec(EQ, y(t), c(n));

{c(n)+(3*n^2+3*n)*c(n+1), c(0) = 0, c(1) = _t[2]}

> 

 


 

Download powerseries.mw

 

Workbooks and page break error...

dharr 8205
July 16 2017
0 1

There is a problem having the page breaks in the file "The Lattice Package Users Guide.mw" - if you remove them then you can load it into a workbook.

I figured this out by assembling the workbook by hand, and it might be instructive to do this. I would install a package from the cloud ("Orbitals" would be my choice but I'm biased) and look at the workbook structure to figure things out.

To make it a package in the cloud just save the workbook to the cloud from the file menu. Then you can install it from the cloud.

The help page ?worksheet,cloud,creatingPackages is quite good on the details of the process

solve using 2nd BC...

dharr 8205
April 05 2017
2 3
ode:=diff(y(x),x$2)+lambda*y(x)=0;
bc:=y(0)=0;
sol:=rhs(dsolve({ode,bc},y(x)));
solve(eval(sol,x=L),lambda,allsolutions);

gives  Pi^2*_Z1~^2/L^2 and then about(_Z1) tells you it is an integer.

Answer for loopless case...

dharr 8205
April 05 2017
0 9

(Edit: Didn't see Carl's solution until after I submitted this.)

There are n*(n-1) = 64 {0,1} adjacency matrices with zeroes on the main diagonal (loopless graphs), so for this case, an indexing function for generating the matrices can help:
 

> 

restart;with(GraphTheory):with(ListTools):
> 

perm:='perm':
n:=3;
bits:=n*(n-1);
indexfn:=proc(i,j) local m;
         m:=(i-1)*n+(j-1);
         if i=j then 0 else perm[m-iquo(m,n+1)] end if;
         end proc:
Matrix(n,n,indexfn);
mlist:=table():
for i from 0 to 2^bits-1 do
   perm:=Bits:-Split(i,'bits'=bits);
   mlist[i]:=Matrix(n,n,indexfn);
end do:
mlist:=convert(mlist,list):
> 

 

n := 3

  

bits := 6

  

Matrix([[0, perm[1], perm[2]], [perm[3], 0, perm[4]], [perm[5], perm[6], 0]])

 (1)
> 

cats:=[ListTools:-Categorize((i,j)->IsIsomorphic(Digraph(i),Digraph(j)),mlist)];

cats := [[Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})]]

 (2)
> 

nops(cats);

16

 (3)
> 

nops(Flatten(cats));

64

 (4)
> 

 

 

Download graphs.mw

Strategy...

dharr 8205
April 05 2017
0 0

If the bugs go away, and you generate a list of the digraphs, then you can categorize into the groups (equivalence classes) that are isomorphic using

[ListTools:-Categorize(IsIsomorphic,graphlist)]

matrix and graph generation...

dharr 8205
April 05 2017
0 0

I would do the generation of the matrices and graphs like this (for the 2x2 case)

> 

n:=2;
bits:=n^2;
mlist:=table():
for i from 0 to 2^bits-1 do
   mlist[i]:=Matrix(n,n,Bits:-Split(i,'bits'=bits));
end do:
mlist:=convert(mlist,list);

n := 2

  

bits := 4

  

mlist := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 1, (2, 2) = 1})]

 (1)
> 

graphs:=map(GraphTheory:-Digraph,mlist):
> 

 


 

Download graphs.mw

indexed procedures...

dharr 8205
March 04 2017
0 1

You can set up indices with procedures, but you stll need parentheses to show it is a procedure.

> 

q := proc () local inds; if not type(procname, 'indexed') then return 'procname(args)' end if; inds := op(procname); inds end proc;

proc () local inds; if not type(procname, 'indexed') then return 'procname(args)' end if; inds := op(procname); inds end proc

 (1)
> 

q(y)

q(y)

 (2)
> 

q[4, 5]()

4, 5

 (3)
> 

q[4, 5]

q[4, 5]

 (4)
> 

``

 

Flawed logic...

dharr 8205
February 17 2017
0 20

If r=0 then you correctly change i. But then you test r agoinst N+1, which fails, and then the else clause sets i back to r, which is zero.

Loop weight...

dharr 8205
February 17 2017
0 1

The Digraph command doesn't seem to allow loops, but adding a weight in a diagonal entry of the Adjacency matrix seems to be accepted, and the weight is given by DrawGraph near the vertex.


 

> 

with(GraphTheory):
> 

A := Matrix([[2,2,0],[0,0,0],[1,3,0]]);

A := Matrix(3, 3, {(1, 1) = 2, (1, 2) = 2, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 3, (3, 3) = 0})

 (1)
> 

G := Graph(A);

GRAPHLN(directed, weighted, [1, 2, 3], Array(%id = 18446744074240237686), `GRAPHLN/table/20`, Matrix(%id = 18446744074240237566))

 (2)
> 

DrawGraph(G);

 
> 

Digraph({[1,1]});

Error, (in GraphTheory:-Graph) invalid edge/arc: [1, 1]

  
> 

 


 

Download Loopgraph.mw

Make some assumptions...

dharr 8205
February 11 2017
0 1

If you add "assuming M>0, g>0;" then you get undefined. If your further choose a limit from the left or right you get -infinity or +infinity.

Give the list of the new variables...

dharr 8205
February 11 2017
0 16

Maple needs to know the new varables are u and v, not W and Q. Try:

Change(V, {x = u-W, y = v-Q}, [u, v]);

Another way...

dharr 8205
December 06 2016
0 2

if Row(A,2).Row(A,2)=0 then...

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