Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 74

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( α

,τ , β ,τ < α ,τ , β ,τ

α

,τ , β ,τ < α ,τ , β ,τ ).

&

The non-strict relation “inclusion about matrix-dimension and index values” is

A ∗ (T ) ⊆

i
v

B ∗ (T ) K ∗ (T ) = P ∗ (T )) L ∗ (T ) = Q ∗ (T )) iff ( & (

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )

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( α

,τ , β ,τ ≤ α

,τ , β ,τ

& α

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,τ , β ,τ ).

The strict relation “inclusion about index values” is

A ∗ (T ) ⊂

i

B ∗ (T ) K ∗ (T ) ⊂ P ∗ (T )) L ∗ (T ) ⊂ Q ∗ (T )))

iff ((( & (

K ∗ (T ) ⊆ P ∗ (T )) L ∗ (T ) ⊂ Q ∗ (T ))) ∨ (( & (

K ∗ (T ) ⊂ P ∗ (T )) L ∗ (T ) ⊆ ) ∗ (T ))) ∨ (( & ( Q & ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )

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i

k
i

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i

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( α

,τ , β ,τ < α ,τ , β ,τ

& α

,τ , β ,τ < α ,τ , β ,τ ).

The non-strict relation “inclusion about index values” is

A ∗ (T ) ⊆

i

B ∗ (T ) K ∗ (T ) ⊆ P ∗ (T )) L ∗ (T ) ⊆ Q ∗ (T ))

( (

iff &

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )

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k k l l

( α

i ,τ , β i ,τ ≤ α

,τ , β ,τ

& α

i ,τ , β i ,τ ≤ α

,τ , β ,τ ).

ETIFIM A ∗ (T ) has temporal strictly increasing elements ,if

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

( μ k i , l j ,τ 1 , ν k i , l j ,τ 1 < ρ k i , l j ,τ 2 , σ k i , l j ,τ 2 ).

ETIFIM A ∗ (T ) has temporal non-strictly increasing elements ,if

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

( μ k i , l j ,τ 1 , ν k i , l j ,τ 1 ≤ ρ k i , l j ,τ 2 , σ k i , l j ,τ 2 ).

ETIFIM A ∗ (T )

has temporal strictly increasing indices, if

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