Alberto Saa

We investigate a growing-network model that combines preferential and uniform attachment with two distinct mechanisms of edge deletion. In addition to the usual uniform random edge deletion, we introduce a novel node-based rule in which uniformly chosen non-isolated nodes lose one of their incident edges. This mechanism differs fundamentally from uniform edge deletion and leads to a nonlinear evolution for the stationary degree distribution due to the nonlinear dependence on the fraction of isolated nodes. We solve the general problem in the stationary regime and obtain closed-form expressions for the degree distribution in terms of hypergeometric and confluent hypergeometric functions. Depending on the balance between attachment and deletion rates, three asymptotic regimes for the degree distribution arise: power-law, exponential, and a critical regime characterized by a stretched-exponential decay. We show that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion. Moreover, we also demonstrate that the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched-exponential decay. Large-scale numerical simulations exhibit excellent agreement with the theoretical predictions.

We propose a Robinson–Trautman evolution in ( 2 + 1 ) -dimensional spacetime that retains key structural features of the four-dimensional case. We consider a recently studied exact family of metrics to select a nonstationary geometry with a cosmological constant, sourced by a null fluid. The metric is completely determined by a single positive function P ( u , ϕ ) , while the corresponding matter content is encoded in a null-fluid density. Motivated by the role of the area-preserving Calabi flow in four dimensions, we introduce a fourth-order length-preserving evolution equation for P ( u , ϕ ) whose stationary configurations correspond, for negative cosmological constant, to boosted BTZ black holes. Numerical solutions strongly support the relaxation of generic regular initial data P ( 0 , ϕ ) toward the stationary sector. The resulting system provides a simple toy model for dissipative dynamics driven by null radiation in lower-dimensional gravity, with several structural similarities to phenomena associated with genuine gravitational radiation.

We investigate a growing network model that combines preferential and uniform attachment with two distinct mechanisms of edge deletion. In addition to the usual uniform probability edge deletion, we introduce a novel node-based rule in which uniformly chosen non-isolated nodes lose one of their incident edges. This mechanism differs fundamentally from uniform edge deletion and leads to a nonlinear evolution for the stationary degree distribution due to the nonlinear dependence on the fraction of isolated nodes. We solve the general problem in the stationary regime and obtain closed-form expressions for the degree distribution in terms of hypergeometric and confluent hypergeometric functions. Depending on the balance between attachment and deletion rates, three asymptotic regimes for the degree distribution arise: power-law, exponential, and a critical regime characterized by a stretched exponential decay. We show that the node-based edge deletion mechanism is less likely to disrupt the scale-free (power-law) regime than the uniform edge deletion. Moreover, we also demonstrate that the precise balance between preferential attachment and node-based deletion can transform a scale-free network into a critical one with stretched exponential decay. Extensive numerical simulations exhibit excellent agreement with the theoretical predictions.

We briefly review the analytical continuation method for determining quasinormal modes (QNMs) and the associated frequencies in open systems. We explore two exactly solvable cases based on the Pöschl–Teller potential to show that the analytical continuation method cannot determine the full set of QNMs and frequencies of a given problem starting from the associated bound state problem in quantum mechanics. The root of the problem is that many QNMs are the analytically continued counterparts of solutions that do not belong to the domain where the associated Schrödinger operator is self-adjoint, challenging the application of the method for determining full sets of QNMs. We illustrate these problems through the physically relevant case of BTZ black holes, where the natural domain of the problem is the negative real line.

We revisit in this note the Hénon’s isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the Hénon’s parabolae and get all isochrone central potentials under mild smoothness assumptions on the potential function. Our approach also allows us to conclude that isochronous radial periods with explicit energy dependence are necessarily Keplerian, i.e., T² ∝ | E|⁻³, and that their corresponding orbits can be easily integrated by mapping them into the usual Kepler problem. It can also be employed to study some other inverse central-force problems and, in particular, it provides a proof of Bertrand’s theorem.