By regularizing the singularities appearing in the two dimensional Regge calculus by means of a segment of a sphere or pseudo-sphere and then taking the regulator to zero, we obtain a simple formula for the gauge volume which appears in the functional integral. Such a formula is an analytic function of the opening of the conic singularity in the interval from π to 4π and in the continuum limit it goes over to the correct result.Much interest has been recently devoted to the discrete approach to two dimensional gravity both in the dynamical triangulation and in the Regge approach [1,2]. It appears that dealing with such a problem in the Regge framework needs a correct treatment of the measure [3]. The attitude we shall adopt here [4] is that to consider the Regge skeleton as defining a geometry described by a metric g µν , which is flat except at the vertices where the curvature becomes singular.The functional integration has to be performed on the metric, with a distance defined by the De Witt super-metric. In order to do that, a gauge fixing has to be introduced and one has to keep into account the correct gauge volume given by the Faddeev-Popov (FP) determinant.Such an approach is the one which has proven successful in the continuum formulation [5][6][7] and has been advocated e.g. by Jevicki and Ninomiya [8] in the Regge framework. In practice the functional integration is given by the integral over the link lengths l i multiplied by a determinant which takes into account how the gauge fixed metric depends on the l i , times the FP determinant. The computation of the first determinant is rather straightforward as it does not involve divergent quantities. Instead we shall concentrate here on the computation of the FP part. It will be shown that following a procedure developed by Aurell and Salomonson [9] for the computation of the determinant of the scalar Laplacian, it is possible to give an exact expression for such a determinant. In the continuum limit it will reproduce the well known result. The FP determinant in question is given by det ′ (L † L) where L is the differential operator which takes from a vector field ξ µ which generates a diffeomorphism to the traceless 1 part of the change in the metric and L † is its adjoint. In general if we denote byĝ µν the background metric and with σ the conformal factor with g = e 2σĝ , we have L = e 2σL e −2σ and
We show that log-periodic power-law (LPPL) functions are intrinsically very hard to fit to time series. This comes from their sloppiness, the squared residuals depending very much on some combinations of parameters and very little on other ones. The time of singularity that is supposed to give an estimate of the day of the crash belongs to the latter category. We discuss in detail why and how the fitting procedure must take into account the sloppy nature of this kind of model. We then test the reliability of LPPLs on synthetic AR(1) data replicating the Hang Seng 1987 crash and show that even this case is borderline regarding the predictability of the divergence time. We finally argue that current methods used to estimate a probabilistic time window for the divergence time are likely to be over-optimistic.
We consider families of geometries of D-dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent of the gauge fixed metric used for describing the geometries. The measure is also invariant in form under an arbitrary change of parameters describing the geometries. We prove the existence of geometries for which there are no related gauge fixing surfaces orthogonal to the gauge fibers. The additional functional integration on the conformal factor makes the measure independent of the free parameter intervening in the De Witt metric. The determinants appearing in the measure are mathematically well defined even though technically difficult to compute.
By adopting the standard definition of diffeomorphisms for a Regge surface we give an exact expression of the Liouville action both for the sphere and the torus topology in the discretized case. The results are obtained in a general way by choosing the unique self-adjoint extension of the Lichnerowicz operator satisfying the Riemann-Roch relation. We also give the explicit form of the integration measure for the conformal factor. For the sphere topology the theory is exactly invariant under the SL(2, C) transformations, while for the torus topology we have exact translational and modular invariance. In the continuum limit the results flow into the well known expressions.
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