Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 75

p
i

p
i

q
j

q
j

k
i

k
i

l l

j

( α , β < α

,τ 2 , β ,τ 2

α , β < α

,τ 2 , β ,τ 2 ).

&

,τ ,τ j ,τ ,τ 1 1 1 1

ETIFIM A ∗ (T )

has temporal non-strictly increasing indices, if

( ∀ τ 1 , τ 2 ∈ T )((τ 1 < τ 2 ) → ( ∀ k i ∈ K )( ∀ l j ∈ L )

p
i

p
i

q
j

q
j

k
i

k
i

l l

j

( α , β ≤ α

,τ 2 , β ,τ 2

α , β ≤ α

,τ 2 , β ,τ 2 ).

&

,τ ,τ j ,τ ,τ 1 1 1 1

4.3 Specific Operations Over ETIFIMs

Let operations ◦ a n d ∗ are d ua l operations of operations ◦ and ∗ , respectively. For

example, pairs ( ◦ , ◦ ) and ( ∗ , ∗ ) can be any a mong pairs ( max , min ) , ( min , max ) .If

◦ and ∗ are average operations, then ◦ and ∗ are also average operations.

Let the time-scale T be fixed, let the ETIFIM

A ∗ (T ) =[ K ∗ (T ), L ∗ (T ), { μ k i , l j ,τ , ν k i , l j ,τ }]

⎧
⎨

⎫
⎬

l
1

l
1

l
n

l
n

l 1 , α

,τ , β ,τ

... l n , α

,τ , β ,τ

k
1

k
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
m

k
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

be given and let k 0 ∈ K and l 0 ∈ L be two indices. Now, we introduce the following

operations over it:

( ◦ , ∗ ) - row-aggregation

A ∗ (T ), ρ ( ◦ , ∗ ) ( k 0 )

⎧
⎨

l
1

l
1

l 1 , α

,τ , β ,τ

...

=

k k

⎩ k 0 , ◦

m α

i ,τ ,

◦

m β

i ,τ

∗

m μ k i , l 1 ,τ ,

∗

m ν k i , l 1 ,τ ...

1 ≤ i ≤ 1 ≤ i ≤ 1 ≤ i ≤ 1 ≤ i ≤

⎫
⎬

l n l n

... l n , α

,τ , β ,τ

| τ ∈ T ,

... ∗

m μ k i , l n ,τ ,

∗

m ν k i , l n ,τ

⎭

1 ≤ i ≤ 1 ≤ i ≤

( ◦ , ∗ )

- column-aggregation

A ∗ (T ),

σ ( ◦ , ∗ ) (

l 0 )