### Table 2. Work factors of optimized classical attacks

### Table 2. Work factors of optimized classical attacks

### Table 1 Variations of Classical Refinement6. Search Work Solution Length

1996

Cited by 31

### Table I Comparison of Commonly Used Engine Efficiencies to Their Equivilent Work Potential FoMs. Component Classical Efficiency Work Potential Equivalent

2001

Cited by 4

### TABLE IV COMPARISONS WITH PREVIOUS WORK USING CLASSICAL HIGH-LEVEL SYNTHESIS BENCHMARKS. NA = RESULTS ARE NOT AVAILABLE.

2004

Cited by 11

### Table 7: Detailed Comparison of Thrust Work Potential and Classical Growth Factors for the F-5E Subsonic Area Intercept Mission.

"... In PAGE 21: ... Consequently, Table 6 is an apples and oranges comparison, but it is useful to illustrate the relationships between the classical methods and those proffered herein. A much more detailed comparison of growth factor results for the F-5E is shown in Table7 . This table shows a group-by group comparison of the growth factor for each component.... ..."

### Table 1. Work factors of classical attacks These results are obtained from the formulas of appendix A. PartIII Improvement of the attacks

"... In PAGE 7: ... The parameters proposed in [6] for this algorithmare p = 2 or p = 3 according to the available memory, and ` = ln k. 7 E ciency of these algorithms Table1 gives the work factor of each of these algorithms (i.e.... ..."

### Table 1. Work factors of classical attacks These results are obtained from the formulas of appendix A. PartIII Improvement of the attacks

"... In PAGE 9: ... The parameters proposed in [6] for this algorithmare p = 2 or p = 3 according to the available memory, and ` = ln k. 7 E ciency of these algorithms Table1 gives the work factor of each of these algorithms (i.e.... ..."

### Table 2 presents informations on explicit results or on the structure of the recurrence relation satis ed by Cm(n) connecting essentially Classical Orthogonal Polynomials. Many results are already published, in print or in preparation. We refer mainly to works related to the recurrence relation approach in which, of course, other approaches and results are mentioned.

"... In PAGE 9: ...139 Pn(x) Qm(x) R[Cm+p(n); ; Cm?q(n)] = 0 xn (J,B,L,He) max 3 terms, solved, [39, 23] x[n] (H,H-E,,K,M,C)) max 3 terms, solved, [39] (J,B,L,He) (J,B,L,He) max 3 terms, partly solved, [9] (H,H-E,K,M,C)) 8 lt; : (H,H-E,K,M,C) x([m] max 3 terms, partly solved, [31, 3, 21] max 3 terms, partly solved, [39] 8 lt; : Pn(x + b); Pn(ax) Pn = (J,B,L,He) (J,B,L,He) 8 lt; : Qm(x) Pm(x) max 5 terms, partly solved, [9] 8 lt; : Pn(x + y); Pn = (H,H-E,K,M,C) (H,H-E,K,M,C) 8 lt; : Qm(x) Pm(x) max 3 terms, partly solved, [31, 3] Bernstein basis - continuous - discrete shifted Jacobi Hahn polynomial 5 terms, solved, [33] (H-J,H-B,H-L,H-He) (J,B,L,He) no mixture max 3 terms, in progress 8 gt; gt; gt; gt; gt; lt; gt; gt; gt; gt; gt; : PiPj Pi; Pj = (J,B,L,He) Pi; Pj = (H,H-E,K,M,C) (J,B,L,He) (H,H-E,K,M,C) 8 lt; : JJ,BB,LL,HeHe 3 terms [28, 20] mixture, max 5 terms, [30] no mixture, max 3 terms, [6] 8 gt; gt; gt; lt; gt; gt; gt; : [Pi]N Pi = (J,B,L,He) Pi = (H,H-E,K,M,C) (J,B,L,He) (H,H-E,K,M,C) in progress (N = 2, max 5 terms) [28] in progress (N = 2, max 9 terms) [22] 8 gt; gt; gt; gt; gt; gt; lt; gt; gt; gt; gt; gt; gt; gt; : P (r) n ; r associated Pn = (J,B,L,He) Pn = (H,H-E,K,M,C) (J,B,L,He) (H,H-E,K,M,C) 8 lt; : r = 1 ; [29, 18, 17] arbitrary r , [27, 18, 17] 8 lt; : r = 1 ; [5, 32] arbitrary r , in progress 8 lt; : (J,B,L,He) (H,H-E,K,M,C) 8 lt; : Dm(J,B,L,He) m(H,H-E,K,M,C) Gram-Charlier series in progress 8 lt; : even part of (J,B,L) odd part of (J,B,L) (J,B,L,He) in progress reciprocal of (J,B,L,He) (J,B,L,He) in progress Table2... ..."

### Table 2: lower bounds for the modality of Borel subgroups in classical groups.

"... In PAGE 4: ... Set r = rank G. Let a be the ideal of b generated by the root spaces relative to the simple roots given in column 3 of Table2 below. Then (a) is a quadratic polynomial in r.... In PAGE 4: ...y Lemma 2.1 we have mod B (a). Thus it su ces to provide the values (a) for the chosen ideals. In Table2 below we list the simple roots whose root spaces generate the ideal a and list (a) for each case. The details of the calculations are omitted.... In PAGE 4: ... (2) With the aid of a computer program written by U. J urgens it was checked for all classical types and r 40, that among all ideals which are generated by root spaces relative to simple roots the ones in Table2 yield maximal values for . (3) It follows from work in [6] that if G is of type Ar for r 9, Br or Dr for r 6, or Cr for r 5, then the bounds given in Table 2 are also upper bounds for mod B.... In PAGE 4: ... J urgens it was checked for all classical types and r 40, that among all ideals which are generated by root spaces relative to simple roots the ones in Table 2 yield maximal values for . (3) It follows from work in [6] that if G is of type Ar for r 9, Br or Dr for r 6, or Cr for r 5, then the bounds given in Table2 are also upper bounds for mod B. Thus we have mod B = (a) in these instances.... In PAGE 4: ... Thus we have mod B = (a) in these instances. We list these cases in Table 3 below together with the ideals a from Table2 . For G of type Ar, for r 7, B3, B4, and C3, the modality of Borel subgroups can also be determined from the information in Table 1 in [4].... In PAGE 7: ...dependent of char K for any of the ideals a in these tables). Therefore, the lower bounds for mod B given in Table2 and thus the ones in Theorem 3.1 in particular do apply in any characteristic, as claimed.... In PAGE 10: ...With the aid of Table2 and Proposition 4.1 we can obtain further re nements of the principal result from [11].... ..."