### Table 2 Symmetries and their associated conservation laws for the SG equations

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### Table 2: The program ConLaw3 applied to compute conservation laws of the sin-Gordon equation.

### Table 2: The program ConLaw3 applied to compute conservation laws of the sine-Gordon equation.

### Table 4: The program ConLaw2 applied to compute conservation laws of the sine-Gordon equation.

### Table 1. Lie apos;s classi cation of invariant second-order ordinary di erential equations No. Equation Symmetry algebra

"... In PAGE 13: ... Surprisingly, it is possible to implement this approach to classifying second-order PDEs (1) by their second-order conditional symmetries in full generality. In Table1 we present the complete list of invariant real second-order ordinary di erential equations together with their maximal invariance algebras, obtained by Lie ([21, 22]). Note that a; k are arbitrary real parameters and f is an arbitrary function.... In PAGE 13: ... Note that a; k are arbitrary real parameters and f is an arbitrary function. As classi cation has been done to within an arbitrary reversible transformation of the variables x; y, the equations given in Table1 are representatives of the conjugacy classes of invariant ordinary di erential equations. Table 1.... In PAGE 14: ...Table1 , since the corresponding ordinary di erential equation is not integrable by quadratures. Next, since our nal aim is to exploit conditional symmetries for the description and reduction of initial value problems, it make no sense to consider case 4.... In PAGE 14: ... Next, since our nal aim is to exploit conditional symmetries for the description and reduction of initial value problems, it make no sense to consider case 4. This is because the symmetry group admitted by the corresponding ordinary di erential equation within the class (18) is the same as that of the more general equation given in case 3 of Table1 . The same argument applies to case 8.... In PAGE 14: ...quation given in case 3 of Table 1. The same argument applies to case 8. Consequently, we will deal only with the remaining cases 2, 3, 5{7, 9. We take as the function in operator (3) the expressions y00 ? f(x; y; y0), where f is one of the right-hand sides of equations listed in the second column of Table1 and make the replacements y ! u, y0 ! ux and y00 ! uxx. We classify PDEs of the form ut = uxx + F (t; x; u; ux) (27) admitting the corresponding Lie-Backlund vector elds.... ..."

### Table 3: The program ConLaw4 applied to compute conservation laws of the sine-Gordon equation.

### Table 4 Implications of Collective Action Clauses for Borrowing Costs

2000

"... In PAGE 11: ... The key variable is UK (versus US) law.11 In Table4 the coefficients on UK law for all four credit-rating categories are significant at standard confidence levels; they 9 To save space, we do not report the coefficients on the selection or issuance equation, having done so previously (Eichengreen and Mody, 1998, 2000). In addition to the country and global variables in the spreads equation, the additional explanatory variables for the issuance equation were debt service relative to exports, short-term debt relative to total commercial bank debt, reserves relative to imports.... In PAGE 14: ...00 0.00 Number of observations 5,309 2,050 Number of bonds 1,781 1,128 Notes: Other than the interaction terms reported here, the specification used is the same as in Table4 . The other coefficients are not reported to conserve space.... ..."

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### Table 2: Local Conservation Laws of (4.1)

2006

"... In PAGE 12: ... For arbitrary functions F (u) and G(u), one has two conservation laws (V1) and (V2); for the case G 0 = F , there are two additional conservation laws (B1) and (B2); for the case G = u, there are also two additional conservation laws (C1) and (C2). The classiflcation is presented in Table2 . [Note that the case where G is linear in u and F = const is the linear case and hence is not considered.... ..."

### Table 5: Spatiotemporal symmetry of bifurcating solutions in nonHopf bifurcation from a discrete rotating wave with ? = O(2), = D 2`, = D `, where ` is odd. The entries for bif are given only up to conjugacy. All bifurcations are period preserving. All bifurcations are pitchforks of discrete rotating waves unless stated otherwise. Notation: MRW = modulated rotating wave, j0 = gcd(j; `). The interpretations in physical space for the cases V0; are identical to those in Example 3.9. The Vj representations, 1 j lt; `=2 lead only to discrete rotating waves due to the presence of re ections in the spatial or spatiotemporal symmetries. However, in the case Vj, L = ?1, there are two branches of bifurcating relative periodic solutions, and one of these has no purely spatial re ection symmetry. For this branch of solutions, the constant term of the _ apos; equation vanishes, but

1998

"... In PAGE 20: ... Recall that V0; denotes the one dimensional irreducible representations of D 2`, and Vj, 1 j lt; `=2 denotes the two dimensional irreducible representations. Our results are almost identical to those given in [25, Table 1] and are tabulated in Table5 . We note that the rst ve columns and most of the eighth column can be read o directly from [25, Table 1].... ..."

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### Table D1 Present values of average costs of conservation and dissemination in perpetuity

1999