19 By comparing (137) and (140b) with (72), then suffi- cient conditions to fulfill (61) in either case ct ′ max = 0 and ct ′ coll = 0 are simply (133). As with the case ct ′ bb = 0, it is not problematic to work out these cases in solving (102) or (103) numerically. As in the case t ′ bb = 0, these cases are easier to handle analytically with the conven- tional variables, since for a given choice of M and E , we have y$_{i}$ = | E | 3 / $^{2}$/M , and thus ct bb follows directly from (136) or (140a). Of course, the three free parameters must comply with the Hellaby–Lake conditions (11). D. Mixed hyperbolic/elliptic configurations. For analytic or qualitative work, the limit k$_{qi}$ → 0 must be handled carefully [24]. While regularity conditions for these mixed models follow also from (60), even if δ ( k ) i → −∞ as k$_{qi}$ → 0, we cannot simply set k$_{qi}$ = 0 in (60). Instead, we examine Γ at constant values of r as r → r$_{b}$ (or k$_{qi}$ → 0). The following series expansions around k$_{qi}$ = 0 hold in this limit A particularly interesting mixed configuration is an el- liptic region surrounded by a hyperbolic exterior given by the choice k$_{qi}$    > 0 for 0 ≤ r < r$_{b}$, elliptic region = 0 for r = r$_{b}$, interface < 0 for r > r$_{b}$, hyperbolic region . (141) ( k ) The fact that δ i → −∞ if k$_{qi}$ → 0 (from its definition, see Appendix A 4), and thus δ ( k ) → −∞ for all t , signals a potential problem in using the quasi–local variables to study an elliptic/hyperbolic model. However, we notice that δ ( k ) does not appear in (102) nor in (103). Also, δ ( k ) i only appears in the initial conditions (108) in the form k$_{qi}$δ ( k ) i , which does not diverge as k$_{qi}$ → 0. Hence, this behavior of δ ( k ) i has no consequences for the numerical integration of these evolution equations for these config- urations. LTB models admit more than one kinematic class in their full radial domain. This type of “mixed” configu- rations can be constructed either by smoothly matching LTB regions with one kinematic class to regions of an- other (as in [24]), or simply by choosing the initial value function k$_{qi}$ (which determines the kinematic class) so that it changes sign in its radial domain. to constants. They have been examined in [10], hence the reader can compare the treatment of these cases in this reference with our treatment in this subsection. The condition for t ′ max = 0 in elliptic models follows directly from (57) as where γ$_{0}$ is a positive constant and x$_{i}$, y$_{i}$ and Z$_{e}$ are given by (54) and (56). As in the case t ′ bb = 0, we can only prescribe one of m$_{qi}$ and k$_{qi}$ , with the other one obtained by solving numerically the constraint (134). If m$_{qi}$ is prescribed, then a necessary condition for the existence of a solution for k$_{qi}$ follows directly from (128b) and (134): ct max = ct$_{i}$ + γ$_{0}$, π − Z$_{e}$ ( x$_{i}$ ) = γ$_{0}$y$_{i}$, (134) 0 < γ$_{0}$k 3 / 2 qi < πm$_{qi}$, (135) which looks like (129) and also does not appear to be restrictive at all. Still, it is necessary to work (134) nu- merically or to make further assumptions on m$_{qi}$ or k$_{qi}$ to get more information. The locus of the big–bang and col- lapse singularities are now given by inserting (134) into (57) ct bb = ct$_{i}$ + γ$_{0}$ − π y$_{i}$ , ct coll = ct$_{i}$ + γ$_{0}$ + π y$_{i}$ , (136) which indicates a time symmetric location of t bb and t coll with respect to t max . The gradients of t bb and t coll are where we used (56) and (31). If t ′ coll = 0, then (57) implies the following constraint similar to (134) ct ′ bb 3 R ′ $_{i}$/R$_{i}$ = π y$_{i}$ ( 3 2 δ ( k ) i − δ ( m ) i ) , ct ′ coll 3 R ′ $_{i}$/R$_{i}$ = π y$_{i}$ δ ( m ) i − 3 2 δ ( k ) i . (137) where ϵ$_{0}$ is a positive constant. Again, we can only pre- scribe one of m$_{qi}$ and k$_{qi}$ , and obtain the other by solving (138). From (128b) we have π < 2 π − Z$_{e}$ ( x$_{i}$ ) < 2 π , hence, if we prescribe m$_{qi}$ a necessary condition for finding k$_{qi}$ as a solution of (138) is ct coll = ct$_{i}$ + ϵ$_{0}$, 2 π − Z$_{e}$ ( x$_{i}$ ) = ϵ$_{0}$y$_{i}$, (138) π < ϵ$_{0}$ k 3 / 2 qi m$_{qi}$ < 2 π, (139) ct bb = ct$_{i}$ + ϵ$_{0}$ − 2 π y$_{i}$ , ct max = ct$_{i}$ + ϵ$_{0}$ − π y$_{i}$ , (140a) ct ′ bb 3 R ′ $_{i}$/R$_{i}$ = 2 π y$_{i}$ ( 3 2 δ ( k ) i − δ ( m ) i ) , ct ′ max 3 R ′ $_{i}$/R$_{i}$ = π y$_{i}$ δ ( m ) i − 3 2 δ ( k ) i , (140b) so that Γ in (60) takes the form which, again, does not seem to be restrictive, though fur- ther information requires either numerical work on (138) or making assumptions on m$_{qi}$ or k$_{qi}$ . The big–bang and maximal expansion times and their gradients are H q c ( t − t$_{i}$ ) ≈ 2 3 ( 1 − 1 L 3 / 2 ) + O ( k$_{qi}$ ) , (142a) 1 − H q qi ≈ 1 − 1 L 3 / 2 + O ( k$_{qi}$ ) , (142b) Γ ≈ 1 + ( 1 − 1 L 3 / 2 ) δ ( m ) i + O ( k$_{qi}$ ) , (143)