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3.2 Mood Vector Space
If we include the scalar multiplication/division by a real number and the sub-
traction to the Mood space set
M
, it represents a vector space. Thus, we have
denoted the elements of
as vectors.
We want to create a vector space because of the interesting properties that we
can implement, such as dynamics, tendencies or distances. In many cases, the
representation of the emotions are well interpreted as “vectors”, some kind of
impulse with certain intensity that moves a person to do something, with some
decay latency.
M
3.3 Extended Mood Space
An Extended Mood Space
M
is an algebraic structure
M
=(
M, ⊕, ,·
),
where (
) is a Mood Space, presenting also the properties for being a Mood
Vector Space (respecting the
M, ⊕
⊕
operator and real number multiplication). If we
add the
operator as an inner product operator and the
·
operator as the
norm,
is considered a normed vector space.
The norm, of course, enables us to calculate distances between two points in
the MVS space. Usually, the distance between emotions or moods is necessary for
the correct identification of the behaviour to use given a sequence of perceived
event (requirement
R3
).
M
3.4 Topological Mood Space
The existence of a normed vector space
M
with the properties presented in
Section 3.3, together with
operator, and the properties
given by this norm operator, provide the possibility to define a topological field
K
operation and the
·
, based on the element addition
⊕
and the scalar multiplication. If
M
is a
vector space over a topological field
is a topological vector space. Indeed,
all normed vector spaces are topological vector spaces. Additionally, if the inner
product
K
,
M
satisfies the properties of (1) symmetry over the product, (2) linearity
with respect the product and (3) linearity with respect the addition, they make
the inner product complete over the field
, thus the Topological Mood Space
defined with this operation is a Hilbert space.
R
3.5 Attenuated Mood Space
The dynamics of emotions requires the inclusion of a mechanism to express their
decay or their transition to a basic temperament position in the PAD space.
Therefore we define an Attenuated Mood Space
M
as an algebraic structure
M
=(
M, ⊕, ,·,A
), where (
M, ⊕, ,·
) is an Extended Mood Space
:
v ∈
and
A
is a family of functions indexed by
M
denoted as
A
=
{a
v
∀
v ∈M
M}
=
{a
v
}
,forwhich
is possible to define an infinite sequence:
v ∈M
−
u
n
:
−
u
n
∈M
n ∈
N
such as:
v
