### Table 1: Max- ow min-cut upper bounds and generalized block Markov lower bounds on C(P; P ) and CFD(P; P ).

2006

"... In PAGE 3: ... In the following discussion we summarize our main results and provide an outline for the rest of the paper. Bounds on capacity: In Section 2, we use the max- ow min-cut upper bound [4] and the generalized block Markov lower bound [4, 5] on the capacity of the relay channel to derive upper and lower bounds on the capacity of the general and FD-AWGN relay channels (see Table1 ). The bounds are not tight for the general AWGN model for any a; b gt; 0 and are tight only for a restricted range of these parameters for the FD-AWGN model.... In PAGE 6: ...that R(P; P ) is in fact achievable by evaluating the mutual information terms in (2) using a jointly Gaussian (U; X; X1). We now show that the lower bound in (2) with the power constraints is upper bounded by R(P; P ) in Table1 . It is easy to verify that I(X; X1; Y ) C (1 + b2 + 2b p )P N ; where is the correlation coe cient between X and X1.... In PAGE 7: ...I(U; Y1jX1) + I(X; Y jX1; U) 1 2 log a2(1 2)P + N N + 1 2 log P + N a2P + N C P N : For a gt; 1, note that h(Y1jX1; U) = h(aX+Z1jX1; U) = h(aX+ZjX1; U) h(X+ZjX1; U) = h(Y jX1; U) and hence I(U; Y1jX1) + I(X; Y jX1; U) C a2(1 2)P N : Note that the above bounds are achieved by choosing (U; X1; X) to be jointly Gaussian with zero mean and appropriately chosen covariance matrix. Performing the maximization over gives the lower bound result in Table1 . This completes the derivation of the lower bound.... In PAGE 9: ... First note that the minimum energy-per-bit for the direct channel, given by 2N ln 2, is an upper bound on the minimum energy-per-bit for both relay channel models considered. Using Theorem 1 and the bounds on capacity given in Table1 , we obtain the lower and upper bounds on the minimum energy-per-bit... In PAGE 10: ...e., 1 + a2 + b2 (1 + a2)(1 + b2) Eb 2N ln 2 min 1; a2 + b2 a2(1 + b2) : To prove the lower bound we use the upper bound ^ C(P; P ) on capacity in Table1 and the relationship of Theorem 1 to obtain the bound. Substituting the upper bound given in Table 1 and taking limits as P ! 0, we obtain the expression Eb 2N ln 2 min 8 gt; lt; gt; : min 0 lt;a2 b2 (1 + )(1 + a2) abp + p1 + a2 b2 2 ; min a2 b2 1 + 1 + a2 9 gt; = gt; ; : To complete the derivation of the lower bound, we analytically perform the minimization.... In PAGE 10: ...hannel given in table, i.e., 1 + a2 + b2 (1 + a2)(1 + b2) Eb 2N ln 2 min 1; a2 + b2 a2(1 + b2) : To prove the lower bound we use the upper bound ^ C(P; P ) on capacity in Table 1 and the relationship of Theorem 1 to obtain the bound. Substituting the upper bound given in Table1 and taking limits as P ! 0, we obtain the expression Eb 2N ln 2 min 8 gt; lt; gt; : min 0 lt;a2 b2 (1 + )(1 + a2) abp + p1 + a2 b2 2 ; min a2 b2 1 + 1 + a2 9 gt; = gt; ; : To complete the derivation of the lower bound, we analytically perform the minimization. For a2=b2, it is easy to see that the minimum is achieved by making as small as possible, i.... In PAGE 10: ... Now we turn our attention to upper bounds on minimum energy-per-bit. Using the lower bound on capacity given in Table1 and the relationship in Theorem 1, we can obtain an... In PAGE 11: ...Table1 satis es the conditions on C(P; P ) in Lemma 1, and therefore, the best upper bound is given by Eb inf 0 lim P!0 (1 + )P R(P; P ): Now we show that this bound gives Eb 2N ln 2 min 1; a2 + b2 a2(1 + b2) : Substituting the lower bound R(P; P ) from Table 1 in theorem 1 and taking the limit as P ! 0, for a gt; 1 we obtain Eb 2N ln 2 min 8 gt; lt; gt; : min 0 lt;a2 1 b2 (1 + )a2 bp(a2 1) + pa2 b2 2 ; min a2 1 b2 1 + a2 9 gt; = gt; ; : To evaluate this bound we use the same approach we used in evaluating the lower bound. We consider the two cases lt; (a2 1)=b2 and (a2 1)=b2 and nd that the minimization is achieved for = (a2 1)b2=(b4 + 2b2 + a2) lt; (a2 1)=b2, and the bound is given by the expression in the theorem.... In PAGE 11: ...the conditions on C(P; P ) in Lemma 1, and therefore, the best upper bound is given by Eb inf 0 lim P!0 (1 + )P R(P; P ): Now we show that this bound gives Eb 2N ln 2 min 1; a2 + b2 a2(1 + b2) : Substituting the lower bound R(P; P ) from Table1 in theorem 1 and taking the limit as P ! 0, for a gt; 1 we obtain Eb 2N ln 2 min 8 gt; lt; gt; : min 0 lt;a2 1 b2 (1 + )a2 bp(a2 1) + pa2 b2 2 ; min a2 1 b2 1 + a2 9 gt; = gt; ; : To evaluate this bound we use the same approach we used in evaluating the lower bound. We consider the two cases lt; (a2 1)=b2 and (a2 1)=b2 and nd that the minimization is achieved for = (a2 1)b2=(b4 + 2b2 + a2) lt; (a2 1)=b2, and the bound is given by the expression in the theorem.... In PAGE 11: ... The new de nition is E(n) = 1 nRn maxk E(n)(k) + E(n) r : It is easy to see that the bounds in this section hold with b replaced by b=p . 4 Side-Information Lower Bounds The lower bounds on capacity given in Table1 are based on the generalized block Markov encoding scheme. In this scheme, the relay node is either required to fully decode the message transmitted by the sender or is not used at all.... In PAGE 28: ... Note that I(X; Y1; YD; YRjX1) = I(X; Y1jX1) + I(X; YDjX1; Y1) + I(X; YRjX1; Y1; YD) = I(X; Y1jX1) + I(X; YDjX1; Y1) = h(Y1jX1) h(Y1jX; X1) + h(YDjX1; Y1) h(YDjX; X1; Y1) h(Y1jX1) + h(YDjX1; Y1) log 2 eN = h(Y1jX1) + h(YDjY1) log 2 eN 1 2 log 2 eVar(Y1jX1) + 1 2 log 2 eVar(YDjY1) log 2 eN = C a2P (1 2) N + C P a2P + N : Similarly, it can be shown that I(X; X1; YD; YR) = I(X; YD; YR) + I(X1; YD; YRjX) = I(X; YD) + I(X; YRjYD) + I(X1; YRjX) + I(X1; YDjX; YR) = I(X; YD) + I(X; YRjYD) + I(X1; YRjX) C P N + C b2 2NP (b2 P (1 2) + N)(P + N) + C b2 P (1 2) N : Again both terms are maximized for = 0. As a result the following upper bound on capacity can be established C min C P N + C b2 P N ; C (1 + a2)P N : Upper and lower bounds in Table1 can be readily established.... ..."

Cited by 6

### Table V lists the number of rows in a redundant parity-check matrix needed to achieve a given stopping distance, for matrices constructed by the generalized HT method, the upper bound from [10], and a bound obtained from cyclic parity-check matrices. In addition, the upper bound from [2], given in Equation (10), is also shown. All results pertain to the [127, 113, 5] BCH code.

708

### Table 1. Lower and Upper Bounds

"... In PAGE 5: ... The box bound and axis bound cnn are not really generalized by the cnn problem, because with respect to this problem their server is constrained; however, since they somehow constitute simpler problems, a dotted line is shown in the figure. Table1 reports the known lower and upper bounds on the compet- itive ratio of the problems, together with the algorithm used to prove the upper bound. 4 Open problems As mentioned above, recently the cnn problem was proved to admit a com- petitive algorithm [17].... ..."

### Table 3.1: Choice of lower and upper bounds for to achieve a maximal absolute di erence of 0 between the generalized logistic cdf and logistic cdf 8

2000

"... In PAGE 3: ...eighborhood around = 0. In addition, they are location and scale invariant (cf. Czado (1997)). Example 1: (Age of Menarche in Warsaw Girls) Milicer and Szczotka (1966) analyzed the occurrence of menarche as a function of age (see Table3 of Stukel (1988) for data). The standard logistic analysis with age as covariate reveals lack of t in the left tail.... ..."

Cited by 4

### Table 2. 4 Conclusion We proved that large compactness is needed for optimal interval routing on certain regular and symmetric topologies used in parallel architectures. The main question remains whether this phenomenon holds also for nearly-optimal interval routing on these topologies. We also improved a lower bound on compactness for the dilation bounded interval routing on general n-vertex graphs2. The complementary upper bound shows that for interval routing with the dilation d1:5De the compactness is O(pn log n). The main unresolved problem is to exhibit a tight trade-o between the dilation and the compactness for general graphs.

1996

"... In PAGE 19: ... 3.5 Summary Table2 summarizes results from Section 3 on bounds for the number of intervals needed for IRS with the bounded dilation.... ..."

Cited by 11

### Table 3: Notation for upper bounds analysis.

2000

"... In PAGE 4: ... Monien and Speckenmeyer showed an upper bound for general CNF formulas, as a function of the number of propositional variables in the formula #28n#29 and the maximum number of literals in any clause #28k#29 #5BMS85#5D. It was based on a version of DPLL, enhanced by autarky analysis, and is of the form: T k #28n#29 #14 P#28L#29 #0D k n #281#29 Notation is explained in Table3 . The parameter #0D k is between 1 and 2, and satis#0Ces the equation #0D k,1 = #0D k,2 + #0D k,3 + :::+1= #0D k,1 , 1 #0D , 1 #282#29 For k =3,#0D 3 =1:618 :::, the #5Cgolden ratio quot;.... ..."

Cited by 2

### Table 1. lower bound upper bound

"... In PAGE 6: ...best attainable in steady state at current altitude within speed limits Roll angle arbitrary Ground speed rate best available given altitude, airspeed, and flight path angle Flight path angle rate lift coefficient amp; structural load factor Roll rate best available at current airspeed Table1 - Performance Limitations Summary The performance limitations, in general, need only be active for a particular pseudo-aircraft when that vehicle can be perceived by the subject. This has the potential to improve situation robustness.... ..."

### Table 1: Upper bound results.

1997

"... In PAGE 3: ... We show that for a speci c choice of and , we get an algorithm which is (4 + p10)=3 lt; 2:38743-competitive for all m 2. For small m, we derive values of and which result in better bounds, summarized in Table1 . Further, we show that for this family of algorithms a lower bound of 2:38518 holds.... In PAGE 3: ...igure 1. Our scheme is di erent from that of Bartal et al. in that they x = 1. Combining this scheme with a preemptive algorithm presented in the next section we get the rst algorithm for PMSR. For 3 m 10, we use the values of and given in Table1 . These values approximate the... In PAGE 8: ... For any xed m, we can bound the values of L1, L2 and L3 by way of a computer program. Using the values of and given in Table1 we have done exactly that for 3 m 10. This method of analysis has proven to be quite useful in that it allowed the author to conjecture the general form of the solution for large m.... ..."

Cited by 16

### Table 5: Parameters of generalized Gray image of extended Hensel lift of QR(n) (left) and minimum distance of the best known linear codes of same length and size (right).

2003

"... In PAGE 6: ... Following this work, we lifted generating polynomial of QR(n) for n = 17, 23, 31, 47 to Z8 and Z16, extended the resulting codes by a parity-check sym- bol, then computed their minimum distance with computer assistance. Results are shown in Table5 , bold (resp. italic) is for codes better than (resp.... ..."

Cited by 1