<s>
In	O
mathematics	O
,	O
the	O
Lyapunov	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
reduction	I-Algorithm
or	O
Lyapunov	O
–	O
Schmidt	O
construction	O
is	O
used	O
to	O
study	O
solutions	O
to	O
nonlinear	O
equations	O
in	O
the	O
case	O
when	O
the	O
implicit	O
function	O
theorem	O
does	O
not	O
work	O
.	O
</s>
<s>
Let	O
be	O
the	O
projection	B-Algorithm
operator	I-Algorithm
onto	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
last	O
equation	O
is	O
now	O
finite-dimensional	O
,	O
since	O
the	O
range	B-Algorithm
of	O
is	O
finite-dimensional	O
.	O
</s>
<s>
Lyapunov	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
reduction	I-Algorithm
has	O
been	O
used	O
in	O
economics	O
,	O
natural	O
sciences	O
,	O
and	O
engineering	O
often	O
in	O
combination	O
with	O
bifurcation	O
theory	O
,	O
perturbation	O
theory	O
,	O
and	O
regularization	O
.	O
</s>
<s>
LS	O
reduction	O
is	O
often	O
used	O
to	O
rigorously	O
regularize	O
partial	O
differential	O
equation	O
models	O
in	O
chemical	O
engineering	O
resulting	O
in	O
models	O
that	O
are	O
easier	O
to	O
simulate	O
numerically	B-Algorithm
but	O
still	O
retain	O
all	O
the	O
parameters	O
of	O
the	O
original	O
model	O
.	O
</s>
