Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 68

⎧
⎨

⎫
⎬

l
1

l
1

l n l n

l 1 , α

,τ , β ,τ

...

l n , α

,τ , β ,τ

1 1

k 1 , α

,τ , β ,τ

μ k 1 , l 1 ,τ , ν k 1 , l 1 ,τ ... μ k 1 , l n ,τ , ν k 1 , l n ,τ

. . .

. . .

≡ | τ ∈ T ,

k
i

k
i

⎩ ⎭

k i , α

,τ , β ,τ

μ k i , l 1 ,τ , ν k i , l 1 ,τ ... μ k i , l n ,τ , ν k i , l n ,τ

. . .

. . .

k m m

k m , α

,τ , β ,τ μ k m , l 1 ,τ , ν k m , l 1 ,τ ... μ k m , l n ,τ , ν k m , l n ,τ

where for every 1 ≤ i ≤ m , 1 ≤ j ≤ n :

μ k i , l j ,τ , ν k i , l j ,τ , μ k i , l j ,τ + ν k i , l j ,τ ∈[ 0 , 1 ] ,

k
i

k
i

k
i

k
i

α

,τ , β ,τ , α ,τ + β ,τ ∈[

0 , 1 ] ,

l
j

l
j

l l

j

α

,τ , β ,τ , α ,τ + β ,τ ∈[

0 , 1 ]

j

and

K ∗ (T ) ={

k
i

k
i

k i , α

,τ , β ,τ |

k i ∈ K & τ ∈ T }

k
i

k
i

={ k i , α

,τ , β ,τ |

1 ≤ i ≤ m & τ ∈ T } ,

L ∗ (T ) ={

l
j

l
j

l j , α

,τ , β ,τ |

l j ∈ L & τ ∈ T }

l
j

l
j

={ l j , α

,τ , β ,τ |

1 ≤ j ≤ n & τ ∈ T } .

Let

K ∗ (T ) ⊂ P ∗ (T )

( ⊂ ) ( ∀ τ ∈ T )

iff K P &

p
i

p
i

k k

( ∀ k i = p i ∈ K : (α

i ,τ < α

,τ )

& (β

i ,τ > β

,τ )).

K ∗ (T ) ⊆ P ∗ (T ) iff ( K ⊆ P ) & ( ∀ τ ∈ T )

p
i

p
i

k
i

k
i

( ∀ k i = p i ∈ K : (α

,τ ≤ α

,τ )

& (β

,τ ≥ β

,τ )).

We must mention that the new IM is 3-dimensional by analogy with the IM
described in Chap. 6 . The indices for the third dimension of the TIFIM and ETIFIM
are elements of a time-scale T :