### Table 1: Local transformations

"... In PAGE 9: ... Thus, there is no need to construct new test sets by pattern generation after local transformations have been applied. Table1 shows some local transformations which are wide-spread in logic synthesis. The circuit realizations are given in Figure 5 and Appendix A.... In PAGE 16: ...Local Transformations and Path Mappings For the local transformations presented in Table1 the circuit realizations and the path mappings are given in the following gures. For parameterized local transformations the circuit realizations and path mappings are given for xed values.... ..."

### Table 2: E ect of a unitary preprocessing transformation U on the operators representing physical quantities in a system P. The rst three entries apply to both unitary and Hermetian operators; the last entry applies to all pairs of unitary operators. System P System PU

"... In PAGE 9: ...physical quantities and coordinate systems. As we will see, the unitary transform U maps physical quantities in P represented by the operators A; B; C; : : : to new quantities in PU represented by the unitarily equivalent operators e A = U?1A U; e B = U?1B U; e C = U?1C U; : : : (18) Table2 summarizes the primary e ects on the operator representations. Note that while all eigen- functions and content measuring transforms change with U, the relative angles between operators and concepts remain unchanged.... In PAGE 10: ...directly from (20), Table2 , and the corresponding results for time and frequency from the previous section. The e T {covariant, e F{invariant transform Fe T = F e F = FT U = U (22) measures e T ; e T (transformed time) content in signals, while the e F{covariant, e T {invariant transform Fe F = Fe T = FT U = F U (23) measures e F; e F (transformed frequency) content in signals.... ..."

### Table 2: E ect of a unitary preprocessing transformation U on the operators representing physical quantities in a system P. The rst three entries apply to both unitary and Hermitian operators; the last entry applies to all pairs of unitary operators.

"... In PAGE 8: ...The corresponding Hermitian representations undergo identical transformations, from A; B; : : : to e A; e B; : : : Table2 summarizes the primary e ects on these operator representations. Note that while all eigenfunctions and content measuring transforms change with U, the relative angles between operators and concepts remain unchanged.... In PAGE 8: ... Results identical to (17){(19) hold for the Hermitian time and frequency operators: T 7! e T = U?1 T U and F 7! e F = U?1F U such that e T?t = ej2 t e F and e Ff = ej2 f e T . The generalized Fourier transforms for the transformed time and frequency operators follow directly from (18), Table2 , and the corresponding results for time and frequency from the previous section. The e T-covariant, e F-invariant transform IFe T = IFe F = IFT U = U (20) measures e T (transformed time) content in signals, while the e F-covariant, e T-invariant transform IF e F = IFe T = IFTU = IFU (21) measures e F (transformed frequency) content in signals.... In PAGE 14: ... U-Cohen apos;s class distributions are covariant by translation to the operators e T and e F: CUe Ff e Tts ( ; ) = CU e Tt e Ffs ( ; ) = (CUs)( ? t; ? f): (43) Therefore, U-Cohen apos;s class distributions measure not joint time T and frequency F content, but joint e T and e F content. Note that, like T and F, e T and e F remain orthogonal concepts (see (14) and Table2 ). In an e ort to keep our notation clean, we will continue to use the coordinates ( ; ) to represent the e T-e F plane.... ..."

### Table 5: Local Circuit Transformations

1996

"... In PAGE 6: ... Furthermore, if fo is de- clared redundant in the transformed circuit, fo and fk are diagnostically equivalent in the original circuit. Experimental results from Table5 in Section 4 show that local circuit transformation is more efficient than a conven- tional diagnostic test pattern generator tool [14] in iden- tifying diagnostic equivalences. The intuition behind this phenomenon is that the conventional diagnostic test pattern generator [14] has a larger search tree to explore, because it has to prove the redundancy of the other s-a-c input fault of G (f2) before it can conclude that f1 and fo are diag- nostically equivalent.... In PAGE 7: ... It is also important to note that there are no fault pairs aborted; every single pair is identified to be diagnostically equivalent by the two methods. The fault pairs considered in Table5 are the distance-0 fault pairs left after the applica- tion of a complete detection test set. Furthermore, since we implemented local circuit transformations for simple gates with exactly two inputs, only distance-0 fault pairs at two input gates are considered.... In PAGE 7: ... Furthermore, since we implemented local circuit transformations for simple gates with exactly two inputs, only distance-0 fault pairs at two input gates are considered. As seen in Table5 , the local cir- cuit transformation reduces the number of backtracks and the runtime for all of the benchmark circuits. Table 6 shows the result of diagnostic equivalence identi- fication using isomorphicsubcircuits.... ..."

Cited by 13

### TABLE IV. The unitary transformation between the CS and FS basis vectors of orbital sym- metry [42]o, isospin I=0, and spin S=1.

### Table 1. The degeneracies of the subspaces belonging to the four eigenspaces k = 0; 1; 2; 3 of the unitary transformation. The dimension goes from 4 to 12, the position where the new dimension appears is indicated in bold.

"... In PAGE 9: ... Each increase N ! N + 1 adds to the degeneracy of one subspace. In the Table1 , we list the degeneracies and indicate in bold the value where the new dimension appears. The pattern is easily recognized.... ..."

### Table 5: E ect of local transformations upon testability

"... In PAGE 13: ... 4 Experimental Results In this section we examine the e ect of local transformations on RD testability for some ISCAS85 circuits [14] and the combinational parts of some ISCAS89 circuits [15]. In Table5 we give two rows for each circuit. In the rst row we give the statistics for the orig- inal benchmark circuit.... In PAGE 13: ...ALT [16]. We applied the same local transformations which were used in [16] to improve CP testability. The results can be improved if the selection among instances of local transformations which exclude one another is directed by special strategies for RD testability. In the rst column of Table5 the name of the circuit is given. The number of logical paths in the circuit is put in the second column.... In PAGE 13: ... Since the number of non-RD paths has been computed by a heuristic, the values given in column RD-FC are lower bounds on the actual fault coverages. The advantages of local transformations can be seen in Table5 . For CP testability as well as for RD testability the fault coverage is increased by local transformations, i.... ..."

### TABLE 9 Welsh production firms by Unitary Authority. Unitary Authority Number

### Table 7.1: Transforming local variables into fields

2003

Cited by 1