Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 71

β

l
j ,τ ,

if v w = l j ∈ L − P

β w,τ = ,

q
s

β

,τ ,

if t w = q s ∈ Q

and

ϕ t u ,v w ,τ , ψ t u ,v w ,τ =

⎧
⎨
⎩

μ k i , l j ,τ , ν k i , l j ,τ , if t u = k i ∈ K

and v w = l j ∈ L − P − Q

ρ p r , q s ,τ , σ p r , q s ,τ , if t u = p r ∈ P − L − K

and v w = q s ∈ Q

=

◦

P (

min (μ k i , l j ,τ , ρ p r , q s ,τ )),

l j = p r ∈ L ∩

∗

P (

max (ν k i , l j ,τ , σ p r , q s ,τ )) , if t u = k i ∈ K and v w = q s ∈ Q

l j = p r ∈ L ∩

0 , 1 , otherwise

Structural subtraction

A ∗ (T ) B ∗ (T ) =[ T ∗ (T ), V ∗ (T ), { ϕ t u ,v w ,τ , ψ t u ,v w ,τ }] ,

where

T ∗ (T ) = ( ) ∗ (T ) ={

t
u

t
u

K − P t u , α

,τ , β ,τ |

t u ∈ K − P } ,

V ∗ (T ) = ( ) ∗ (T ) ={ v w , α v w,τ , β w,τ | v w ∈ L − Q L − Q } ,

for the set-theoretic subtraction operation and

t
u

k
i

α

,τ = α ,τ ,

for t u = k i ∈ K − P ,

β w,τ = β

l
j

,τ ,

for v w = l j ∈ L − Q

and

ϕ t u ,v w ,τ , ψ t u ,v w ,τ = μ k i , l j ,τ , ν k i , l j ,τ ,

for t u = k i ∈ K − P and v w = l j ∈ L − Q .

Negation of an ETIFIM

(T ) ∗ =[

T ∗ (T ), V ∗ (T ), {¬ μ k i , l j ,τ , ν k i , l j ,τ }] ,

¬

A

where ¬ is one of the negations from Table 2.1 from Sect. 2.1, or another possibly

defined.