### TABLE 4. Eigenvalues up to 305 for 4, the fundamental domain of the group ?4 of Section 3.5. The entries of type D are also the eigenvalues for 4, corresponding to the group E4.

### Table 8. Probabilities of fundamental and contagious defaults assuming zero bankruptcy costs and a complete market structure, i.e. banks diversify their inter-bank business as much as possible. Bank failure scenarios are grouped by the number of fundamental defaults. For each group, the probability that only fundamental and that fundamental and contagious defaults occur are shown. A fundamental default is due to the losses arising from exposures to market risk and credit risk to the corporate sector, while a contagious default is triggered by the default of another bank who cannot fulfill its promises in the inter-bank market.

"... In PAGE 26: ... The initial structure which exploits the structural information about the multi-tier architecture of the Austrian banking system can be characterized as an incomplete market structure. The simulation results with the complete structure and the long run scenario are re- ported in Table8 . Analogous to Table 4, we group bank failure scenarios by the number of fundamental defaults.... ..."

### Table 5 Di erential invariants of transformation groups in C 2 Fundamental Invariant Lie di erential invariant(s) one-form determinant

"... In PAGE 6: ... Apparently, he did not publish the formulas for the di er- ential invariants of the three contact transformation groups. A complete list of di erential invariants and invariant one-forms for the point and contact transformation groups appears in Table5 . In this table, and below, we use the notation un = Dn xu for the higher order derivatives of the scalar function u.... In PAGE 8: ... For example, the invariant third order equations are all of the form uxuxxx = cu2 xx. The full list of Lie determinants for all Lie groups of point and contact transformations in the complex plane can also be found in Table5 . In this table, we have omitted any inessential constant factors.... In PAGE 8: ... In this table, we have omitted any inessential constant factors. Detailed results on the symmetry classi cation of ordinary di erential equations, and their integation, follows immediately from the results in Table5 . See Lie, [14], for a full range of applications.... In PAGE 9: ... In our geometric interpretation, then, ! = ds is the G-invariant element of arc length, and I is and arbitrary function of the curvature and its derivatives with respect to arc length. Table5 immediately provides a symmetry classi cation of the scalar variational prob- lems, generalizing results of Gonz alez{L opez, [8], for point transformation groups. Recall rst that, in the scalar case, a nth order Lagrangian L(x; u(n)) is called nonsingular if it satis es the nondegeneracy condition @2L(@un)2 6 = 0.... ..."

### Table 1. Decompositions of spherical triangles with non-fundamental vertex The decompositions of the sphere are given in stereographic projection. The triangles are represented by the values of their angles. The multiple is omitted. In the most part of cases the group G generated by re ections of p coincide with the group generated by re ections of fundamental triangle f. In the rest cases the triangles fundamental for G are presented above the tables.

1999

Cited by 2

### Table 6. Probability of bank defaults and amount of contagion. Scenarios are grouped by total bank defaults (fundamental and contagious). For each group the probability of that state is shown as well as quantiles of the fraction of contagious defaults. In the long run analysis we assume zero bankruptcy costs, whereas in the short term scenario insolvent banks are assumed to default completely and pay nothing.

"... In PAGE 22: ... It indicates that despite the fact that contagion is a rare event there are situations where it accounts for a major part of defaults. The results in Panel A of Table6 confirm the intuition that the fraction of contagious defaults increases in the number of total defaults. As we have already seen before contagion is a minor problem in the long run case (Panel B of Table 6).... ..."