### Table 1. Old and new bounds for d ! 1 in the unit size l0-model (the classical model). A simpler proof for an old result is denoted by +. The continuous model is new to this paper, whereas all classical models are discrete.

"... In PAGE 2: ...he 4-node uniform cycle. On the other hand, we show that 2.5 is a lower bound when we impose a natural restriction on the way how a request can be served. Table1 shows the best previous bounds along with our new bounds. Note that a bound proved in one model can imply the same bound in another model (e.... ..."

### Table 1. The areas of proof theory, organized by goals.

2001

"... In PAGE 8: ... I first present a very quick overview of the present goals of proof theory. Table1 gives a three-fold view of proof theory, in which proof theory is split into three broad categories based on the goals of the work in proof theory. The first column represents the traditional, classic approaches to mathe- matical proof theory: in this area the goal has been to understand stronger and stronger systems, from second-order logic up through higher set the- ories, and especially to give constructive analyses of the proof-theoretic strengths of strong systems.... ..."

Cited by 3

### Table 1. The areas of proof theory, organized by goals.

2001

"... In PAGE 8: ... I rst present a very quick overview of the present goals of proof theory. Table1 gives a \three-fold quot; view of proof theory, in which proof theory is split into three broad categories based on the goals of the work in proof theory. The rst column represents the traditional, classic approaches to mathe- matical proof theory: in this area the goal has been to understand stronger and stronger systems, from second-order logic up through higher set the- ories, and especially to give constructive analyses of the proof-theoretic strengths of strong systems.... ..."

Cited by 3

### Table 2.Elementary Landscapes. Problem Move Set K 1? NAES Hamming

"... In PAGE 44: ...Grover [57] observed that the landscapes of a number of classical combinatorial optimization prob- lems are of this form, see Table2 . In order to keep the notation consistent with Grover apos;s work for regular graphs we introduce K = K0=D where D is as usual the vertex degree.... ..."

### Table 1: The rules of the classical system KE.

"... In PAGE 15: ... There is (under very plausible assumptions) a unique sys- tem of rules of this kind: the system KE described in Mondadori 1988, D apos;Agostino and Mondadori 1994. Its rules (in the version for signed for- mulae) are given in Table1 . All the rules are tree-expansion rules, like the tableau rules, and the notions of closed tree, refutation and proof are de ned in the same way as for tableaux.... ..."

### 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: Results for Pigeonhole problems 4.4 The Mutilated Checkerboard Problem A classic problem in Arti cial Intelligence is the Mutilated Checkerboard Problem: whether one can cover with dominoes an n n checkerboard from which two squares from opposite corners have been removed [McCarthy, 1964]. Although there is an elegant proof of the impossibility 9

1994

"... In PAGE 9: ...sing the equivalent of Strategy A of Section 4.1). This di erence allowed obdds to solve the problem up to n = 13 in a time comparable to that taken by ldpp for n = 11. Table2 summarizes the results.... ..."

Cited by 43

### lable. Proof:

2000

Cited by 1

### Table 1: Computing times in milliseconds (left) and speed-up ratios (right) with respect to the classical algorithm.

1996

"... In PAGE 9: ... We used the PACLIB environment [3] which combines the com- puter algebra library SACLIB [2] with the parallel features of the System library [1]. Table1 lists computing times and computing time ratios for inputs of various lengths. The row heading 20=15 refers to a dividend of 20 words and a divisor of 15 words.... In PAGE 9: ... The sequential implementation splits the quotient as in the proof of Theorem 1; the parallel implementation splits the quotient as in the proof of Theorem 2. Table 2 has the same structure as Table1 , but instead of the computing time it lists the num- ber of digit products that were computed. Those numbers agree very well with the bounds given in Theorems 1 and 2.... ..."

### Table 2: Count of digit products (left) and ex- pected speed-up ratios (right) with respect to the classical algorithm.

1996

"... In PAGE 9: ... The sequential implementation splits the quotient as in the proof of Theorem 1; the parallel implementation splits the quotient as in the proof of Theorem 2. Table2 has the same structure as Table 1, but instead of the computing time it lists the num- ber of digit products that were computed. Those numbers agree very well with the bounds given in Theorems 1 and 2.... In PAGE 9: ... This can be explained by noting that IQR and Algorithm H use digit divisions in order to determine the quotient di- gits. Table2 counts those digit divisions as digit products, but the true cost of digit division is about 2.5 times the cost of a digit product in the SACLIB implementation we used.... ..."