### Table 1: Useful dualities between hypermatrix and Kronecker algebra expressions.

"... In PAGE 6: ...Nowlet A ,A xy r ;; T ,T r ;; d ,D xy (12) (note that A, T and d are readily constructed from fu r (x)g,fv r (y)g,ft r ( )gg and fd(x;; y;; )g in (5), using relations (12), (9), (10) and (40)). Then, using the duality properties of Table1 in Appendix A, wehave that A I h i , A xy r ;; vec c (T) ,T r 4 Note that this technique is a special case of the alternating orthogonal projections onto convex sets algorithm [28], where the convex sets in our case are actually linear spaces.... In PAGE 20: ... In particular, there exists a one-to-one correspondence between hypermatrix expressions and matrix operations in the Kronecker algebra. We list in Table1 a few dualities instrumentaltoourwork. Symbol \:= quot; is used to specify a hypermatrix assignment.... In PAGE 20: ... Symbol denotes the Kronecker product (de ned below), while byvec c (A)itismeant the stack of the columns of A into a single vector. Note in the third line of Table1 the use of tensor summation notation with... ..."

### Table 1: Useful dualities between hypermatrix and Kronecker algebra expressions.

1997

"... In PAGE 6: ...Now let A ,A xy r ; T ,T #12 r ; d ,D xy#12 #2812#29 #28note that A, T and d are readily constructed from fu r #28x#29g,fv r #28y#29g,ft r #28#12#29gg and fd#28x; y; #12#29g in #285#29, using relations #2812#29, #289#29, #2810#29 and #2840#29#29. Then, using the duality properties of Table1 in Appendix A, wehave that A #0A I h#12i , #16 A xy#12 r #16 #12 ; vec c #28T#29 ,T r #16 #12 4 Note that this technique is a special case of the alternating orthogonal projections onto convex sets algorithm #5B28#5D, where the convex sets in our case are actually linear spaces.... In PAGE 20: ... In particular, there exists a one-to-one correspondence between hypermatrix expressions and matrix operations in the Kronecker algebra. We list in Table1 a few dualities instrumental to our work. Symbol #5C:= quot; is used to specify a hypermatrix assignment.... In PAGE 20: ... Symbol #0A denotes the Kronecker product #28de#0Cned below#29, while byvec c #28A#29 it is meant the stack of the columns of A into a single vector. Note in the third line of Table1 the use of tensor summation notation with... ..."

### Table 2: Some properties of the Kronecker product.

"... In PAGE 7: ... r into standard matrix form. In fact, the minimization task becomes: min T k ; A I h i vec c (T) ; dk 2 The sought for least{squares solution is vec c (T)= ; (A I h i ) T (A I h i ) ;1 (A I h i ) T d (13) = ; (A T I h i )(A I h i ) ;1 (A T I h i )d =(A T A I h i ) ;1 (A T I h i )d = ; (A T A) ;1 I h i ; A T I h i d = ; (A T A) ;1 A T I h i d where wehave used the properties of the Kronecker product summarized in Table2 in Appendix A. Expression (13) contains a Kronecker product with an identity matrix, which produces a (typically) large sparse matrix.... In PAGE 7: ...parse matrix. This drawback can be avoided as follows. Let D ,D xy := D xy . Then, using the last propertyof Table2 , and noting that d =vec c (D T ), one proves that the last identity of (13) is equivalentto T = ; (A T A) ;1 A T D T (14) No Kronecker product is present in equation (14). This last reduction, though, cannot be used for the design of the basis lters when constraints on the length of the single 1-D kernels are imposed, as described in Section 2.... In PAGE 21: ... . a p1 B a pq B 1 C C C A (43) Some properties of the Kronecker product relevanttoourwork are listed in Table2 . It may be noticed that the writing A BC is ambiguous, because (A B)C 6 = A (BC) (actually, either one of such quantities... ..."

### Table 2: Some properties of the Kronecker product.

1997

"... In PAGE 7: ...#12 r into standard matrix form. In fact, the minimization task becomes: min T k , A #0A I h#12i #01 vec c #28T#29 , dk 2 The sought for least#7Bsquares solution is vec c #28T#29= , #28A #0A I h#12i #29 T #28A #0A I h#12i #29 #01 ,1 #28A #0A I h#12i #29 T d #2813#29 = , #28A T #0A I h#12i #29#28A #0A I h#12i #29 #01 ,1 #28A T #0A I h#12i #29d =#28A T A #0A I h#12i #29 ,1 #28A T #0A I h#12i #29d = , #28A T A#29 ,1 #0A I h#12i #01, A T #0A I h#12i #01 d = , #28A T A#29 ,1 A T #0A I h#12i #01 d where wehave used the properties of the Kronecker product summarized in Table2 in Appendix A. Expression #2813#29 contains a Kronecker product with an identity matrix, which produces a #28typically#29 large sparse matrix.... In PAGE 7: ... Let D ,D xy #12 := D xy#12 . Then, using the last propertyof Table2 , and noting that d =vec c #28D T #29, one proves that the last identity of #2813#29 is equivalentto T = , #28A T A#29 ,1 A T D #01 T #2814#29 No Kronecker product is present in equation #2814#29. This last reduction, though, cannot be used for the design of the basis #0Clters when constraints on the length of the single 1-D kernels are imposed, as described in Section 2.... In PAGE 21: ... . a p1 B a pq B 1 C C C A #2843#29 Some properties of the Kronecker product relevant to our work are listed in Table2 . It may be noticed that the writing A#0ABC is ambiguous, because #28A#0AB#29C 6 = A#0A#28BC#29 #28actually, either one of such quantities... ..."

### Table 8: Social relative total factor productivity for different farm size categories* Average Social Total Factor Productivity** Region

in Title Author

"... In PAGE 36: ... However, the point here is to determine to what an extent farm size influences the farmer apos;s ability to capture benefits and use the structure of incentives. The results obtained from the social TFP analyses, which are summarized in Table8 , indicate that: * Average social TFP is lower than average private TFP in all the regions. The difference is much more accentuated at the beginning of the 1 980s than later in the decade when some of the privileges were already removed.... ..."

### Table 1: Useful dualities between hypermatrix and Kronecker algebra expressions. Associativity A (B C) = (A B) C

"... In PAGE 6: ... Now let A , Axy r ; T ,T r ; d , Dxy (12) (note that A, T and d are readily constructed from fur(x)g,fvr(y)g,ftr( )gg and fd(x; y; )g in (5), using relations (12), (9), (10) and (40)). Then, using the duality properties of Table1 in Appendix A, we have that A Ih i , Axy r ; vecc(T) , T r 4Note that this technique is a special case of the alternating orthogonal projections onto convex sets algorithm [28], where the convex sets in our case are actually linear spaces.... In PAGE 20: ... In particular, there exists a one-to-one correspondence between hypermatrix expressions and matrix operations in the Kronecker algebra. We list in Table1 a few dualities instrumental to our work. Symbol \:= quot; is used to specify a hypermatrix assignment.... In PAGE 20: ... Symbol denotes the Kronecker product (de ned below), while by vecc(A) it is meant the stack of the columns of A into a single vector. Note in the third line of Table1 the use of tensor summation notation with... ..."

### Table 1: Useful dualities between hypermatrix and Kronecker algebra expressions. Associativity A (B C) = (A B) C

"... In PAGE 6: ... Now let A , Axy r ; T ,T r ; d , Dxy (12) (note that A, T and d are readily constructed from fur(x)g,fvr(y)g,ftr( )gg and fd(x; y; )g in (5), using relations (12), (9), (10) and (40)). Then, using the duality properties of Table1 in Appendix A, we have that A Ih i , Axy r ; vecc(T) , T r 4Note that this technique is a special case of the alternating orthogonal projections onto convex sets algorithm [28], where the convex sets in our case are actually linear spaces.... In PAGE 20: ... In particular, there exists a one-to-one correspondence between hypermatrix expressions and matrix operations in the Kronecker algebra. We list in Table1 a few dualities instrumental to our work. Symbol \:= quot; is used to specify a hypermatrix assignment.... In PAGE 20: ... Symbol denotes the Kronecker product (de ned below), while by vecc(A) it is meant the stack of the columns of A into a single vector. Note in the third line of Table1 the use of tensor summation notation with... ..."

### Table 6. Rank of western Kentucky wetland sites according to net biomass production compared with possible abiotic and biotic predictors of productivity of forested wetlands

"... In PAGE 14: ... We emphasize that none of the variables used here is completely independent from the others in this sys- tem and, indeed, productivity itself is directly relat- ed to variables such as tree density, biomass, and average age. Few abiotic or biotic measurements, by them- selves, predicted the ranking of productivity of these wetland systems ( Table6 ). Three abiotic vari- ables (percent of flooding in the growing season, average water depth, and total phosphorus in the IWer) predicted the two most productive wetlands (H1 and C1) but did not predict the low productivi- ty of the stagnant wetland (C2).... ..."

### Table 1: Independence numbers of the strong product of three moderate sized odd cycles

"... In PAGE 9: ...Table 1: Independence numbers of the strong product of three moderate sized odd cycles The results presented in Table1 were obtained mostly using backtracking algorithms. For instance, the computations for (C7 2 C7 2 C9) took about one week.... In PAGE 9: ... For instance, the computations for (C7 2 C7 2 C9) took about one week. In [25] it was also conjectured that the independence numbers of C7 2 C7 2 C2k+1 and C7 2 C9 2 C2k+1 reach the upper bound from Table1 if k is large enough. This is indeed true for the rst case, as follows from the main result of this section: Theorem 6 Let k 3.... ..."