Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 70

⎧
⎨

l
j

β

,τ ,

if v w ∈ L − Q

q
s

β w,τ = β

,τ ,

if t w ∈ Q − L ,

⎩ q

s

l
j

min (β

,τ , β

,τ ),

if t

w ∈

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 , if t u = k i ∈ K and v w = l j ∈ L − Q

or t u = k i ∈ K − P and v w = l j ∈ L ;

ρ p r , q s ,τ , σ p r , q s ,τ , if t u = p r ∈ P and v w = q s ∈ Q − L

= or t u = p r ∈ P − K and v w = q s ∈ Q ; ,

◦ (μ k i , l j ,τ , ρ p r , q s ,τ ), if t u = k i = p r ∈ K ∩ P

∗ (ν k i , l j , t , σ p r , q s , t ) , and v w = l j = q s ∈ L ∩ Q

0 , 1 , otherwise

Termwise multiplication

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

where

T ∗ (T ) = K ∗ (T ) ∩ P ∗ (T ) ={

t
u

t
u

t u , α

,τ , β ,τ |

t u ∈ K ∩ P & τ ∈ T } ,

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

p
r

t
u

k
i

α

,τ =

min (α

,τ , α

,τ ),

for t u = k i = p r ∈ K ∩ P ,

q
s

β w,τ =

l
j

max (β

,τ , β

,τ ),

for v w = l j = q s ∈ L ∩ Q

and

ϕ t u ,v w ,τ , ψ t u ,v w ,τ =◦ (μ k i , l j ,τ , ρ p r , q s ,τ ), ∗ (ν k i , l j ,τ , σ p r , q s ,τ ) .

Multiplication

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 − L t u , α

,τ , β ,τ |

t u ∈ K ∪ ( P − L ) } ,

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

α

k
i

,τ ,

if t u = k i ∈ K

t u

α

,τ =

L ,

p
r

α

,τ ,

if t u = p r ∈ P −