Search

We study how bank networks can be driven, via diffusion, to a state where they exhibit greater resistance to a systemic shock. Firstly without making any assumption about the dynamics which drives the interbank lending in the network we prove that the state in which the banks exhibit greater resistance to a systemic shock is the state S in which they have equal leverages. Then we introduce a diffusion–like law which drives the interbank lending dynamics in the network. We prove that the steady state of the system is precisely the state S. In the general case, where the bank network is modelled by a directed, weighted graph, we define two projection operators, one acts on the space of flows of loans and the other one acts on the space of the leverages. The projection operator projects the vector of the initial flows (the vector of the initial leverages) to the vector of the steady state where, in both cases, all the leverages are equal. Examples are given where the networks are driven to their optimal and suboptimal states; when the steady state can only be achieved by reversion of the flows of loans a suboptimal state can be determined. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

We use the conditional symmetry approach to study the r evolution of a minisuperspace spherically symmetric model both at the classical and the quantum level. After integration of the coordinates t, theta and phi in the gravitational plus electromagnetic action the configuration space dependent dynamical variables turn out to correspond to the r-dependent metric functions and the electrostatic field. In the context of the formalism for constrained systems (Dirac-Bergmann, Arnowitt-Deser-Misner) with respect to the radial coordinate r, we set up a pointlike reparametrization invariant Lagrangian. It is seen that, in the constant potential parametrization of the lapse, the corresponding minisuperspace is a Lorentzian three-dimensional flat manifold which obviously admits six Killing vector fields plus a homothetic one. The weakly vanishing r Hamiltonian guarantees that the phase space quantities associated to the six Killing fields are linear holonomic integrals of motion. The homothetic field provides one more rheonomic integral of motion. These seven integrals are shown to comprise the entire classical solution space, i.e. the space-time of a Reissner-Nordstrom black hole, the r-reparametrization invariance since one dependent variable remains unfixed, and the two quadratic relations satisfied by the integration constants. We then quantize the model using as supplementary conditions acting on the wave function, the quantum analogues of the various subalgebras of the classical conditional symmetries. We find that, as a semiclassical analysis shows, in all but one allowed case the ensuing solutions to the Wheeler-DeWitt equation exhibit a good correlation with the classical regime. In the remaining case, the emerging semiclassical geometry is a four-dimensional homogeneous space-time, thus exhibiting no curvature singularity.

A conditional symmetry is defined, in the phase space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry. The use of these symmetries as quantum conditions on the wave function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein-Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at the first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on Ψ. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrization generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler-DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two "scale factors". © 2013 .

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler - DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two “scale factors”.

A proposal is put forward which enables the canonical quantization of a family of spherically symmetric geometries in 3+1 dimensions. The proposal consists of a particular renormalization Assumption and an accompanying Requirement and results in a Wheeler DeWitt equation which is based on a renormalized manifold parametrized by three smooth scalar functionals. The aforementioned equation is analytically solved for the 3+1 case.

The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper imposition of the quantum analogues of two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re-normalization assumption, severely reduces the possible state vectors to three unique (up to general coordinate transformations) smooth scalar functionals. The quantum analogue of the Hamiltonian constraint produces a Wheeler-DeWitt equation based on this reduced manifold of states, which is completely integrated. © 2008 IOP Publishing Ltd.