### Table 7.1 Asymptotic upper and lower bounds on envelope size and work for an overlap graph in d di- mensions.

1997

Cited by 6

### Table 1: Translation table for extremal expressions of the equation x0 i = fi(xi). Let (fi) be the desired bound on x0 i ( = L or = U). The table is applied recursively to the subexpressions of fi. The symbol xj is any state variable other that xi, c is a constant, M+ and M? are monotonic functions, c and c return the lower or upper range values of c, M and M return the lower or upper functional envelope of the monotonic function. For the state variable xi, the bound is the same as that for fi.

1998

"... In PAGE 11: ... Nsim constructs an extremal ODE system from a QDE by minimizing and maximizing each derivative equation in the SQDE. To compute an extremal equation, Nsim rewrites the SQDE using the translations in Table1 to substitute terms. For example, the ODE system A0 = c ? f(A) B0 = f(A) ? f(B) becomes A0 = c ? f(A) B0 = f(A) ? f(B) A0 = c ? f(A) B0 = f(A) ? f(B) [Table 1 about here.... In PAGE 11: ... To compute an extremal equation, Nsim rewrites the SQDE using the translations in Table 1 to substitute terms. For example, the ODE system A0 = c ? f(A) B0 = f(A) ? f(B) becomes A0 = c ? f(A) B0 = f(A) ? f(B) A0 = c ? f(A) B0 = f(A) ? f(B) [ Table1 about here.] By applying both a lower and an upper translation for each equation, Nsim generates a new ODE system whose state variables are the upper and lower bounds for the state variables of the SQDE.... In PAGE 31: ... Part One Let x0 i = fi(x) be the ith ODE in A. Referring to Table1 , we can see that for each of the \ground quot; expressions e = c; xi, or xj we have that L(e) e U(e) whenever Condition 4 holds. By induction on e, we can show that the other expressions in the table also obey L(e) e U(e).... ..."

Cited by 13

### Table 1: Translation table for extremal expressions of the equation x0 i = fi(xi). Let (fi) be the desired bound on x0 i ( = L or = U). The table is applied recursively to the subexpressions of fi. The symbol xj is any state variable other that xi, c is a constant, M+ and M? are monotonic functions, c and c return the lower or upper range values of c, M and M return the lower or upper functional envelope of the monotonic function. For the state variable xi, the bound is the same as that for fi.

1998

"... In PAGE 5: ... Part One Let x0 i = fi(x) be the ith ODE in A. Referring to Table1 , we can see that for each of the \ground quot; expressions e = c; xi, or xj we have that L(e) e U(e) whenever Condition 4 holds. By induction on e, we can show that the other expressions in the table also obey L(e) e U(e).... In PAGE 30: ... Nsim constructs an extremal ODE system from a QDE by minimizing and maximizing each derivative equation in the SQDE. To compute an extremal equation, Nsim rewrites the SQDE using the translations in Table1 to substitute terms. For example, the ODE system A0 = c ? f(A) B0 = f(A) ? f(B)... ..."

Cited by 13

### Table 1 The effect of observability in the two-tank cascade. Each entry represents the ratio of the area of the envelope when selected measurements are made to the area with no measurements. The envelope area is defined to be the integral of the difference between the upper and lower bounds over the domain of interest. The absolute envelope areas for no measurements are 6357, 3764, 782, and 2772 when measured over A 2T0;100U for f.A/, B 2T0;70U for g.B/,andt 2T0;40U for A and B Measured variables Envelope area ratio

1999

"... In PAGE 27: ...x 16/ C 24;15px U for x gt; 16. Table1 shows the results of this test and Fig. 17 (top) shows the static envelopes for f and g when all four variables are measured.... ..."

### Table 2. Modi cations to the envelope models. parameter magnitude notes

"... In PAGE 8: ... To do so, we have computed envelope models which have boundary conditions only at the surface, considering models with the same surface pa- rameters as those given in Table 1. For each of these eight models we have considered the modi cations summarized in Table2 . Here c is the mixing length parameter, X the hydrogen abundance, the opacity, rc the radius at the base of the convection zone and EOS refers to the equation of state; all the models use the EFF formulation (Eggle- Table 2.... In PAGE 8: ... For envelope models it is not possible to compute low-degree p-mode fre- quencies, and hence we cannot obtain G(!) from a t to equation (2). However, as remarked previously, the di er- ences in the function G for the models corresponding to the small modi cations considered in Table2 can be expected to be very similar to those in Gas, which can be computed for envelope models. As an illustrative case, we shall show the results for just one case, an envelope model with the same global and surface parameters as Model 5 in Table 1.... In PAGE 8: ... In Fig. 10, we show the di erences Gf as(!) between the models with the changes indicated in Table2 and the refer- ence model (an envelope model corresponding to Model 5 in Table 1). As expected, Gf as is signi cant for the change in the equation of state or the envelope abundances, as a result of the corresponding changes in ?1 in the second helium ion- ization zone; however, it is also important for modi cations in the atmospheric opacities or the mixing-length param- eter c because these modi cations change, for instance, the depth of the second helium ionization layer.... In PAGE 9: ... The continuous line is Gas(!) for an envelope model similar to Model 5 in Table 1. The dashed line corresponds to the change in Te , the dot-dashed line to the change in the mixing- length parameter c, and the dotted line to the change in the at- mosphericopacity, for the changes listed in Table2 . The functions have been shifted by constants to match at a given frequency.... In PAGE 9: ... Figure 10. Di erencesin the ltered phase-functionGf as between envelope models with the modi cations indicated in Table2 and the reference model (corresponding to Model 5 in Table 1) for: change in Te (continuous line), change in the mixing length pa- rameter c (dotted line), change in the atmospheric opacities (long dashed line), change in X (dot-dashed line), change in Z (short-longdashedline) and changein the equationof state (short dashed line). 4 CONCLUSIONS By analysing low-degree (l 2) p-mode frequencies for the Sun we have shown that it is possible to t an asymptotic expression which allows to separate the contribution of the upper layers as given by a function of frequency G(!) from that of the interior.... ..."

### Table 2. The 1.4 GHz ATCA Radio Source Counts

"... In PAGE 11: ... For these sources we have adopted the peak ux in comput- ing the source counts, while for all the others, lying above the upper envelope, we have adopted the total ux. In Table2 the 1.4 GHz source counts are presented.... ..."

### Table 1. Selected quantities of our solar models (the solar age is taken as 4.57 Gyr): upper table subscripts refer to the initial composition (i), to surface (S), center (C) and to the bottom of the convective envelope (b.c.); neutrino fluxes (lower table) are given in s?1cm?2; the last line gives the predicted flux (in SNU) for the Chlorine and GALLEX experiments.

### Table 1 The envelope contents

2003

"... In PAGE 10: ... The information required by the Finder Agent to look for a piece in Carrel is held in an electronic Sealed Envelope. The information contained in the envelope is summarized in Table1 . The Selection Function is the part of the information contained in the Sealed Envelope that allows the Finder Agent to perform a nego- tiation.... ..."

Cited by 2

### Table 1 The envelope contents

"... In PAGE 9: ... The information required by the Finder Agent to look for a piece in Carrel is held in an electronic Sealed Envelope. The information contained in the envelope is summarized in Table1 . The Selection Function is the part of the information contained in the Sealed Envelope that allows the Finder Agent to perform a negotiation.... ..."