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NullGaussche

NullGaussche is a fictional mathematical construct used in theoretical discussions and in science fiction to explore the interplay between null spaces of linear operators and Gaussian smoothing.

In the fictional framework, let A: V -> W be a linear map between finite-dimensional real vector

Properties: As sigma decreases to zero, K_sigma(A) tends to Kernel(A). As sigma grows, K_sigma(A) expands toward

In simple examples, if A has a one-dimensional kernel spanned by a vector v, then K_sigma(A) approximates

Etymology and usage: The name combines "null" (kernel) and "Gauss" (Gaussian). The term appears in fictional glossaries

spaces.
For
each
sigma
>
0,
define
K_sigma(A)
as
the
subspace
obtained
by
applying
a
Gaussian
blur
of
standard
deviation
sigma
to
the
kernel
of
A,
producing
a
sigma-smoothed
version
of
the
null
space.
The
NullGaussche
of
A,
denoted
NG(A),
is
the
family
{K_sigma(A)
|
sigma
>
0}.
The
concept
is
used
to
study
how
small
perturbations
or
noise
influence
the
reach
of
the
null
space.
a
larger
subspace,
reflecting
increased
smoothing.
The
construction
is
compatible
with
composition
and
with
changes
of
basis,
reflecting
a
form
of
invariance
under
linear
isomorphisms.
NG(A)
provides
a
lens
on
stability
of
solutions
under
Gaussian
perturbations.
span{v}
for
small
sigma,
while
for
large
sigma,
K_sigma(A)
may
become
the
whole
space
in
the
limit
of
heavy
smoothing.
and
in
speculative
writeups
about
linear
algebra.