Information Technology Reference
In-Depth Information
Mathematics should not take the place of text: readers quickly get lost if they need
to decipher a stream of complex expressions.
Let
={
i
=
1
ʱ
i
x
i
|
ʱ
i
∈
=
i
=
1
ʱ
i
x
i
and
S
F
,
1
≤
i
≤
n
}
.For
x
=
i
=
1
ʲ
i
x
i
, so that
x
=
ʱ(
i
=
1
ʱ
i
x
i
)
+
y
,
y
∈
S
,wehave
ʱ
x
+
ʲ
y
ʲ(
i
=
1
ʲ
i
x
i
)
=
i
=
1
(ʱʱ
i
+
ʲʲ
i
)
x
i
∈
S
.
Although the mathematics in this example is straightforward, there is no motivation,
and the thicket of symbols is daunting.
Let
S
be a vector space defined by
F
n
S
=
1
ʱ
i
x
i
|
ʱ
i
∈
.
i
=
We now show that
S
is closed under addition. Consider any two vectors
=
i
=
1
ʱ
i
x
i
and
y
=
i
=
1
ʲ
i
x
i
. For any constants
x
,
y
∈
S
. Then
x
ʱ, ʲ
∈
F
,wehave
n
n
ʱ
x
+
ʲ
y
=
ʱ
1
ʱ
i
x
i
+
ʲ
1
ʲ
i
x
i
i
=
i
=
n
=
1
(ʱʱ
i
+
ʲʲ
i
)
x
i
,
i
=
so that
ʱ
x
+
ʲ
y
∈
S
.
Note the vertical alignment of the equality symbols.
Mathematical expressions should not run together.
For each
x
i
,1
≤
i
≤
n
,
x
i
is positive.
Each
x
i
, where 1
≤
i
≤
n
, is positive.
If a formula is complex, or is a key result, it should be displayed. In such displays, the
formula can be either centred or indented; choose either, but be consistent. However, if
part of the display is an algorithm or program, centering can look peculiar. Displayed
formulas (or graphs or diagrams) should be positive results, not counter-examples,
so that readers who skim through the paper won't be misled. If a displayed formula
is sufficiently important it should be numbered, to allow discussion of it elsewhere
in the paper and for reference once the paper is published. As in the example above,
a displayed formula should be treated as a phrase; and remember to add trailing
punctuation to the formula, with appropriate spacing.
The weighting function can then be simplified to
f
x
+
f
y
+
w
μ
=
(
1
−
μ)
l
x
+
μ
l
y
,
μ
thus showing that
controls the blending.
