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In-Depth Information
a
≤
a
a
≥
a
, then loss of the information within the boundaries of
l
-
If
,
l
3
il
3
l
4
il
4
(
)
st and
l
-th terms is equal to the sum of squares of two triangles.
One triangle has its base equal to
+
1
a
−
a
, and another triangle has its base
il
3
l
3
a
−
a
equal to
, too. Let us determine heights of these triangles.
l
4
il
4
a
−
a
a
−
a
As triangles with bases
and
are similar, we have
il
3
l
3
l
4
il
4
h
a
−
a
⎧
1
=
il
3
l
3
;
⎪
⎨
h
a
−
a
2
l
4
il
4
⎪
⎩
h
+
h
=
1
where
h
and
h
are heights of corresponding triangles. Hence, the height of
triangles with the base
1
1
a
−
a
is equal to
3
3
il
l
2
3
a
a
−
a
a
−
a
a
+
a
a
−
a
a
+
a
+
a
a
−
a
a
h
=
il
3
l
4
il
3
l
3
l
3
il
4
l
3
il
3
l
3
l
4
l
l
3
il
4
l
3
il
3
=
(
)(
)
1
a
−
a
a
−
a
−
a
+
a
l
4
l
3
l
4
l
3
il
4
il
3
a
−
a
=
il
3
l
3
.
a
−
a
−
a
+
a
l
4
l
3
il
4
il
3
a
−
a
Height of triangles with the base
is equal to
l
4
il
4
a
−
a
a
−
a
h
=
1
−
il
3
l
3
=
l
4
il
4
.
1
a
−
a
−
a
+
a
a
−
a
−
a
+
a
4
3
4
3
4
3
4
3
l
l
il
il
l
l
il
il
In this case information loss is equal to
(
)
(
)
2
2
a
−
a
+
a
−
a
l
3
il
3
l
4
il
4
.
(
)
(4.23)
2
a
−
a
+
a
−
a
l
4
il
4
il
3
l
3
a
>
a
a
<
a
If
,
, then information loss is equal to
l
3
il
3
l
4
il
4
(
)
(
)
2
2
a
−
a
+
a
−
a
−
l
3
il
3
l
4
il
4
.
(
)
2
a
−
a
+
a
−
a
(4.24)
Thus, from (4.20) — (4.24) it follows that the general loss of the information
l
4
il
4
il
3
l
3
(
)
(
)
⎧
⎫
⎡
2
2
⎤
1
m
−
1
k
1
a
−
a
+
a
−
a
⎪
⎨
⎪
⎬
∑∑
(
)
σ
δ
1
δ
2
,
=
a
−
a
+
a
−
a
+
l
3
il
3
l
4
il
4
⎢
⎣
⎥
⎦
(
)
il
l
3
il
3
l
4
il
4
il
k
2
2
a
−
a
+
a
−
a
⎪
⎩
⎪
⎭
l
==
11
i
l
4
il
4
il
3
l
3
(4.25)
