Information Technology Reference
In-Depth Information
In general, a vector space over an algebraic field
C
(for example real or complex
numbers) is defined when a set of elements, called
vectors
, are given with an internal
sum operation
which is commutative and associative, and for which a zero vector
0
exists with respect to this sum, and an opposite vector
⊕
−
v
exists for any vector
v
⊕−
=
,
⊕−
−
such that
v
v
0
(for two vectors
v
w
,then
v
w
is often abbreviated by
v
w
).
Moreover, a scalar multiplication is defined between any element
c
∈
C
of the field
and any vector
v
such that
c
v
is a vector too, and some natural axioms hold for this
multiplication (distributivity of scalar multiplication with respect to vector addition
a
·
(
v
⊕
w
)=
av
⊕
aw
, and distributivity of scalar multiplication with respect to field
addition
(
a
+
b
)
·
v
=
a
·
v
⊕
b
·
v
). Finally for two scalars
a
,
b
,if
ab
is their product
in the field
a
·
(
b
·
v
)=(
ab
)
·
v
and 1
·
v
=
v
if 1 is the multiplicative identity of the
field
C
.
In conclusion, we can realize that measures allow us to locate objects and popu-
lations in spaces. In the simplest case this space is the unidimensional real line, but
in general, it is a vector space of many dimensions.
5.4.3
Some of Euler's Jewels
In this topic we encountered some spectacular results due to Leonhard Euler. For
example, the famous identity
e
i
π
+
0 and the infinite sum representations of
e
x
,sin
x
,andcos
x
. In this section we shortly present one of Euler's most sensa-
tional discoveries. It is the solution of a problem, called Basel problem, which was
raised by the Italian mathematician Pietro Mengoli and was investigated also by
Jakob Bernoulli. The question concerns the exact determination of the limit of the
p
-
harmonic series
for
p
1
=
=
2:
∞
1
1
n
p
It was proved that for
p
=
1 the harmonic series is divergent, but for
p
>
1these
series are convergent.
The following is the solution found by Euler in 1735 (since 1731, at the age of
24, he was hardly working on this problem).
According to Euler's determination of the series for sine we have:
x
3
3!
+
x
5
5!
−
x
7
7!
+
x
9
9!
...
=
−
sin
x
x
therefore:
x
8
9!
...
Now the main intuition of the argument can be introduced. We know that given
an algebraic equation
P
x
2
3!
+
x
4
5!
−
x
6
7!
+
sin
x
x
=
1
−
(
x
)=
0ofdegree
n
with non-zero roots
a
1
,
a
2
,...,
a
n
and
P
by means of the following product (which is easily
proved to have the same roots and
P
(
0
)=
1 we can express
P
(
x
)
(
0
)=
1):
