Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 73

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

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

e
v

iff ( & (

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )( μ k i , l j ,τ , ν k i , l j ,τ ≤ ρ k i , l j ,τ , σ k i , l j ,τ ).

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

A ∗ (T ) ⊂

e

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

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

A ∗ (T ) ⊆

e

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

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )( μ k i , l j ,τ , ν k i , l j ,τ ≤ ρ k i , l j ,τ , σ k i , l j ,τ ).

The strict relation “inclusion about matrix-dimension and indices” is

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

i
d

iff ((( & (

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

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

p
i

p
i

q
i

q
i

k
i

k
i

l
i

l
i

( 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 indices” is

A ∗ (T ) ⊆

i
d

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

& ( ∀ τ ∈ (T ))( ∀ k ∈ K )( ∀ l ∈ L )

p
i

p
i

q
i

q
i

k
i

k
i

l
i

l
i

( α

,τ , β ,τ = α ,τ , β ,τ

& α

,τ , β ,τ = α ,τ , β ,τ ).

The 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