<s>
In	O
mathematics	O
,	O
noncommutative	B-Algorithm
residue	I-Algorithm
,	O
defined	O
independently	O
by	O
M	O
.	O
and	O
,	O
is	O
a	O
certain	O
trace	O
on	O
the	O
algebra	O
of	O
pseudodifferential	O
operators	O
on	O
a	O
compact	O
differentiable	O
manifold	O
that	O
is	O
expressed	O
via	O
a	O
local	O
density	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
circle	O
,	O
the	O
noncommutative	B-Algorithm
residue	I-Algorithm
had	O
been	O
studied	O
earlier	O
by	O
M	O
.	O
and	O
Y	O
.	O
in	O
the	O
context	O
of	O
one-dimensional	O
integrable	O
systems	O
.	O
</s>
