### Table 3: the modality of Borel subgroups in classical groups of small rank.

"... In PAGE 4: ... Thus we have mod B = (a) in these instances. We list these cases in Table3 below together with the ideals a from Table 2. 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].... ..."

### Table 1: uniform lower bounds for the modality of Borel subgroups in classical groups.

"... In PAGE 3: ...roup of G. Let r = rank G. There exists a quadratic polynomial f 2 Q[t] such that mod B f(r): That is, the modality of B grows quadratically with the rank of G. More speci cally, depending on the type of G the polynomial f may be taken as in Table1 below.... In PAGE 4: ...Table1 the one for type Ar is minimal (for r 4). Thus, we may formulate a uniform lower bound for mod B independent of the type of G: Corollary 3.... In PAGE 4: ... Then (a) is a quadratic polynomial in r. Moreover, (a) f(r), where f(r) may be taken as in Table1 above. Proof.... In PAGE 4: ... For a xed classical type we choose for f(r) the polynomial (a) which is minimal for that type. This yields the lower bounds of Table1 . Whence, Proposition 3.... In PAGE 4: ...uadratically with the rank of G. Thus the polynomial bounds in Theorem 3.1 are optimal in terms of their degrees. Considering the ratio of mod B by dim Bu as r grows for a xed classical type, we infer from Table1 that for all classical groups 1 6 lim r!1 mod B dim Bu 1: The same lower bound can be derived for type Ar from [7] (second part of the proof of Theorem 3.... In PAGE 4: ...or mod B. Thus we have mod B = (a) in these instances. We list these cases in Table 3 below together with the ideals a from Table 2. 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 Table1 in [4]. At present is not known whether mod B is a polynomial in r as suggested by these results.... In PAGE 6: ... [6]. For G2 this information can also be read o from Table1 in [4]. Type of G a dim a mod B = (a) A5 1; 3; 5 13 1 A6 1; 3; 5 18 1 A7 1; 4; 7 22 2 A8 1; 4; 7 29 3 A9 1; 4; 8 35 4 B3 2 7 1 B4 1; 3 14 2 B5 1; 4 21 3 B6 1; 4 29 5 C3 1; 3 8 1 C4 1; 4 13 2 C5 1; 5 19 3 D4 2 9 1 D5 3 15 2 D6 1; 4 25 4 Table 3: the modality of Borel subgroups in classical groups of small rank.... In PAGE 7: ...roup of G. Let r = rank G and s = rankss P . There exists a quadratic polynomial f 2 Q[t] such that mod P f(r ? s): That is, the modality of P grows at least quadratically with r ? s, the di erence of the semisimple ranks of G and P . Moreover, the polynomial f may be taken from Table1 above. Proof.... In PAGE 8: ...1 and Theorem 3.1, we infer that mod P mod Q f(r ? s) for some f 2 Q[t] from Table1 according to the type of H. We illustrate the procedure in the proof of Theorem 4.... ..."

### 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].... ..."

### TABLE I. Borel sum rule predictions for the ? ! mixing parameters. Parameter Zero Widths Physical Widths

### Table 1: Values for fB and fBs from QCD sum-rules in dependent on the b-quark mass. The Borel parameter window is M2 = (4{8) GeV2.

"... In PAGE 12: ... We then use the same values for mb, s0 and M2 in both the QCD sum-rule for fB and the light-cone sum-rules for the form factors,5 which helps to reduce the systematic uncertainty of the approach. The corresponding parameter- sets and results for the decay constants are given in Table1 . The question of the value of the b-quark mass has attracted considerable attention recently; following these developments [20], we use the value mb = (4:8 0:1) GeV.... In PAGE 16: ...t is a 0.8% e ect. This result means that, as for radiative corrections, there is a strong cancellation of the mb dependence in the ratio of the light-cone correlation function to fB. The same statement holds for the dependence on the continuum-threshold within the limits speci ed in Table1 . For the dependence on the Borel parameter, we nd an 7% e ect, increasing with q2, which again reminds us of the fact that the light-cone sum rules become less reliable for large q2.... ..."

### Table 3. Indicators and Weights Used in the USNWR University Rankings

2005

"... In PAGE 7: ...he others. Therefore, we restrict our analyses to these 129 institutions. The USNWR rankings use seventeen measures considered related to institutional quality. These are shown in Table3 . 9 Only fourteen of these were available on the website.... ..."

### TABLE V ANAESTHESIA EMERGENCE*

### Table 1. The xed point U coordinate u U as a function of n. u U;Pad e: obtained on the basis of the [1/1] Pad e approximant; u U;Res: obtained by the Pad e-Borel resummation. n 1 2 3 4 5 6 7 8

1