### Table 2. Proof: The proof is constructed by formulating an algebraic test for the existence of

2001

"... In PAGE 5: ... This step is performed for the other eight modes to derive the conditions in Table 2. Table 1: Equilibria of the single cell automaton H 1 Mode Equilibrium q 1 v #03 D =0;v #03 N =0 q 2 v #03 D = R D #15 D ;v #03 N =0 q 3 v #03 D =0;v #03 N = R N #15 N q 4 v #03 D = R D #15 D ;v #03 N = R N #15 N q 5 v #03 D = R D #15 D #08 1 2 , mh D #09 ;v #03 N =0 q 6 v #03 D = R D #15 D ;v #03 N = R N #15 N #08 m#28u N , h N #29+ 1 2 #09 q 7 v #03 D = R D #15 D #08 1 2 , m , R N #15 N + h D #01#09 ;v #03 N = R N #15 N q 8 v #03 D =0;v #03 N = R N #15 N #08 m #28u N , h N #29+ 1 2 #09 q 9 v #03 D = R D #15 D #02 1 2 , m #08 R N #15 N , m #28u N , h N #29+ 1 2 #01 + h D #09#03 ;v #03 N = R N #15 N #08 m#28u N , h N #29+ 1 2 #09 Table2 : Existence conditions for equilibrium points of H 1 Mode Existence condition q 1 0 #3Ch D , 1 2m ^ u N #3Ch N , 1 2m q 2 0 #15 h D + 1 2m ^ u N #3Ch N , 1 2m q 3 , R N #15 N #3Ch D , 1 2m ^ u N #15 h N + 1 2m q 4 , R N #15 N #15 h D + 1 2m ^ u N #15 h N + 1 2m q 5 h D , 1 2m #14 0 #3Ch D + 1 2m ^ u N #3Ch N , 1 2m q 6 , R N #15 N #08 m #28u N , h N #29+ 1 2 #09 #15 h D + 1 2m ^ h N , 1 2m #14 u N #3Ch N + 1 2m q 7 h D , 1 2m #14, R N #15 N #3Ch D + 1 2m ^ u N #15 h N + 1 2m q 8 , R N #15 N #08 m #28u N , h N #29+ 1 2 #09 #3Ch D , 1 2m ^ h N , 1 2m #14 u N #3Ch N + 1 2m q 9 h D , 1 2m #14, R N #15 N #08 m#28u N , h N #29+ 1 2 #09 #3Ch D + 1 2m ^ h N , 1 2m #14 u N #3Ch N + 1 2m The existence conditions of the equilibria are visualized in Figure 7. The axes of the plot are the Delta threshold h D and the external input u N biased with the Notch threshold h N .... In PAGE 5: ... The axes of the plot are the Delta threshold h D and the external input u N biased with the Notch threshold h N . The inequality constraints from Table2 translate to half spaces in the #28u N , h N ;h D #29 plane and the regions in which a particular modal equilibrium exists is the intersection of the half spaces for that particular mode. For example, mode q 9 is contained in the slant-hatched parallelogram at the center of Figure 7, whose two vertical sides are the half spaces h N , 1 2m #14 u N and u N #3Ch N + 1 2m , and two oblique sides are the half spaces h D , 1 2m #14, R N #15 N n m #28u N , h N #29+ 1 2 o and , R N #15 N n m #28u N , h N #29+ 1 2 o #3Ch D + 1 2m .... In PAGE 6: ... Proposition 3 #28Selection of switching thresholds#29 For the two cell hybrid automaton H 2 , there exist only three discrete states containing equilibria: , q 1 2 ;q 2 3 #01 , , q 1 3 ;q 2 2 #01 and , q 1 9 ;q 2 9 #01 , if the thresholds h D and h N are chosen to be: , 1 2m #3E h D #3E 1 2m , R M #15 M R D #15 D , 1 2m #3E h N #3E 1 2m #284#29 Proof: The existence condition of discrete states , q 1 2 ;q 2 3 #01 and , q 1 3 ;q 2 2 #01 are identical because of symmetry and is given as the inequality constraints #284#29. As previously pointed out #28also refer to Table2 and Figure 7#29 the constraints on h D given in the constraint set #284#29 eliminate the equilibria in all discrete states of H 1 other than q 2 , q 3 , q 6 , q 8 and q 9 . Hence the only possible equilibria of H 2 are contained in the pairwise combinations of the above #0Cve discrete states.... ..."

Cited by 2

### TABLE I TEST VECTOR PAIRS USED IN THE PROOF OF THEOREM 1.

2003

### TABLE I TEST VECTOR PAIRS USED IN THE PROOF OF THEOREM 1.

2003

### Table 2: The Metropolitan Museum of Art Proof of Concept Test: Terms Per Participant, for those who tagged 80% of the works.

2006

"... In PAGE 10: ...Table2... ..."

Cited by 3

### Table 1: Diagnosis Results for STG3 Test Vectors Circuit Test Faults Candidate List Size Execution Time (s) PROOFS PROOFS RAPSIM

1996

"... In PAGE 5: ... All experi- ments were performedon a SUN SPARCstation 20 workstation with 64MB of memory for the ISCAS89 benchmark circuits [13]. Table1 shows the diagnosis results for stuck-at faults. In the case of real faulty circuits the failing responses are ob- tained from a tester.... ..."

Cited by 5

### Table V. Target, Proof, and Printed Values for Test Colors on Images Shown in Figures 14b and 18; the Values Are Chromaticity and Luminance (x, y), Y

1988

Cited by 35

### Table10 = test(m; in(e; k; s)): 4. Very similar to the previous proof.

1994

"... In PAGE 40: ... If eq(m; n) = f then test(n; rem(n; in(e; m; s))) = test(n; in(e; m; rem(n; s))). Using Table10 and eq(m; n) = f, we see that the r.... ..."

Cited by 3

### Table 1 gives some of the example systems with which we tested our proof plan. PT stands for the total elapsed planning time, given in seconds. Space constraints do not allow us to provide a full description of each veri cation problem. These example veri cation problems are all taken from [7]. They all involve the use of generalise, UFI and equation solving. The success rate of the proof plan was

"... In PAGE 11: ... Table1 . Some example veri cation conjectures 83%, with an average total elapsed planning time of 750 seconds, and standard deviation of 345.... ..."

### Table 1. Benchmark of recording the multiplier proof (Time in seconds and size in bytes)

1995

"... In PAGE 13: ... It contains all the sub-proofs in the corresponding ML le. Table1 lists the time taken to record this proof and the proof le size. Two tests was carried out: the rst with the proof recorder disabled; and the second with it enabled.... In PAGE 13: ... This extra time is acceptable since the proof les will only be generated after the proof is completed satisfactorily (probably once) and be ran in batch mode. The sizes of the proof les are also listed in Table1 . They are very large (43 Table 1.... ..."

Cited by 12

### Table 27: Proof of Proposition 5.1

2002

"... In PAGE 27: ... Hence, we obtain the table. Table27 : It is known that determining whether the conjunction of two FBDD formulas fi1 and fi2 is consistent is NP-complete (Gergov amp; Meinel, 1994b) Moreover, FBDD satisfies :C. Since fi1 ^ fi2 is inconsistent iff fi1 j= :fi2, we can reduce the consistency test into an entailment test.... ..."

Cited by 59