Bias Element and Calculus errata

[jacko...@yahoo.com]

d^n(f)/dy^n != d^n(f)/dx^n*(dx/dy)^n

but enough of that

program
k = 1 bit
m = a bits
n = 1 bits
p = reversable pseudo random number sequence
p(x) = x bits of the sequence

function forward
q=p(2);
r=p(1);
s=p(1);
switch(q)
case(00): if(n==0) sample of k valid else m++;
case(01): if(k==0) m++ carry into n;
case(1X): if(k==1) m++ carry into n;
endswitch;
if(n and q=00) begin
k^=r;
if(k==0) m++ no carry;
end;
end forward

i think has a very good chance of being biased w.r.t. k
the dual clicks of 1X as opposed to 01, providing for quicker inversion
of k if k is 0 ?

i do wonder about polarities and problem inversions.....

[jacko...@yahoo.com]

mr christchofel's gamma?

[jacko...@yahoo.com]

cleaning this up to

function forward
q=p(2);
r=p(1);
s=p(1);
switch(q)

case(00): if(n==0) sample of k valid;

case(01): if(k==0) m++ carry into n;
case(1X): if(k==1) m++ carry into n;
endswitch;

if(n and q=00) k^=r;
end forward

one could point out that this is of no use, and you may have been
right. the critical point is when m is just about to carry into n and
make n==0 again. this is called the injection point. with 0.5 chance of
1 being injected for sampling and 0.25 chance of 0 being injected for
sampling. and 0.25 chance of being 50:50 randomized for the next
injection attempt. this leads to twice as many ones being injected, but
1 is sampled only half as much and so the effects cancel and 50:50
sampling bias is produced which is quite useless, blah, blah

dont say i told you so just yet, this code below uses a reinjection
pricipal.

function forward
q=p(2);
r=p(1);
s=p(2);
switch(q)
case(00): if(n==0) sample of k valid;
case(01): if(k==0) begin
/* a strange mod 2 effect and 3/8 of injectable zeros recycled */
m++ carry into n;
if(s!=00 and n) m++ no carry;
end;

case(1X): if(k==1) m++ carry into n;
endswitch;

if(n and q=00) k^=r;
end;
end forward

this makes the 0.25 injection of zero become a (0.25)*5/8 injection
probability, and recycles the randomization process. after calculation
of the probability series, an 16/5:1 ratio of 1s to 0 s are injected,
which at a 2:1 sample rate gives 8/5:1 overall sampled bias. all steps
in forward are reversable as all operations are non destructive and the
if conditions are not altered by the then actions.

is this ok???

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
s=p(2);
switch(q)
case(00): if(n==0) sample of k valid;

case(01): if(k==0) m++ carry into n;
case(1X): if(k==1) m++ carry into n;
endswitch;

if(n) begin
k^=r;
/*recycle if ready to recycle (datum end points if statement)*/
if(k==0 and s!=00) m++ no carry;
end;
end forward

is this ok??

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
s=p(2);
switch(q)
case(00): if(n==0) sample of k valid;
case(01): if(k==0) m++ carry into n;
case(1X): if(k==1) m++ carry into n;
endswitch;
if(n) begin
k^=r;
/*recycle if ready to recycle (datum end points if statement)*/

if(k==0 and s!=00 and m=-1) m=0;

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
s=p(4);
l=n;
switch(q)
case(00): if(n==0) sample of k valid else {if(s==-1) k^=r;};
case(01): if(k^n==0) m++ carry into n;
case(1X): if(k^n==1) m++ carry into n;
endswitch;
if(l!=n) k=!k;
end forward

is this a perfect symetry or what?

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
s=p(1);

t=!s;
switch(q)
case(s0): if(n==0 and s==0) sample of k valid; if(n==1 and s==1) k^=r;
case(s1): if(k==0) m++ carry into n;
case(tX): if(k==1) m++ carry into n;
endswitch;
end forward

a dimension of asymetry added and the join problem is eliminated.

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
s=p(1);
t=!s;
switch(q)

case(00): if(n==0 and s==0) sample of k valid; if(n==1 and s==1) k^=r;
case(01): if(k==s) m++ carry into n;
case(1X): if(k==t) m++ carry into n;
endswitch;
end forward

the final design?

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);
v=p(1);
t=!s;
switch(q)
/* asymetric process */

case(00): if(n==0 and s==0) sample of k valid; if(n==1 and s==1) k^=r;

/* odd or even exchange */
case(00): if(n!=s and v==0) begin n=!n; s=!s; end;
/* two to one motion bias */

case(01): if(k==s) m++ carry into n;
case(1X): if(k==t) m++ carry into n;
endswitch;

end forward;

seems like connection most probable between poles of s dimension.

[jacko...@yahoo.com]

function forward
q=p(2);
r=p(1);

switch(q)
/* asymetric process */
case(00): if(n==0 and s==0) sample of k valid; if(n==1 and s==1) k^=r;
/* odd or even exchange */

case(00): if(n!=s) begin n=!n; s=!s; t=!s; end;