Black hole solutions in gravity with nonminimal derivative coupling and nonlinear material fields

Scalar–tensor theory of gravity with nonlinear electromagnetic field, minimally coupled to gravity is considered and static black hole solutions are obtained. Namely, power-law and Born–Infeld nonlinear Lagrangians for the electromagnetic field are examined. Since the cosmological constant is taken into account, it allowed us to investigate the so-called topological black holes. Black hole thermodynamics is studied, in particular temperature of the black holes is calculated and examined and the first law of thermodynamics is obtained with help of Wald’s approach.

Second-Order Multiplier Iteration Based on a Class of Nonlinear Lagrangians

Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

A CLASS OF NONLINEAR LAGRANGIANS: THEORY AND ALGORITHM

This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

TOPOLOGY CHANGE AND SIGNATURE CHANGE IN NON-LINEAR FIRST-ORDER GRAVITY

We show that different topologies of a space-time manifold and different signatures of its metric can be encompassed into a single Lagrangian formalism, provided one adopts the first-order (Palatini) formulation and relies on nonlinear Lagrangians, that were earlier shown to produce, in the generic case, universality of Einstein field equations and of Komar's energy-momentum complex as well. An example in Relativistic Cosmology is provided.

A Model of Affine Gravity in Two Dimensions and Plurality of Topology

A new model of two-dimensional gravity with an action depending only on a linear connection is suggested. This model is a topological one, in the sense that the classical action does not contain a metric or zweibein at all. A metric and an additional vector field are instead generated in the process of solving the equations of motion for the connection. The general solution of these equations of motion is given by an arbitrary Weyl connection which can be described by using the space of orbits under the action of the conformal group in the functional space containing all pairs formed by a metric and a vector field. By choosing a gauge one obtains a constant curvature equation. It is shown that this model admits an equivalent description by using a family of Lagrangians depending on the metric and the connection as independent variables. We show that nonlinear Lagrangians in the first order formalism lead to plurality of topology for the manifolds under consideration and give a simple general mechanism of governing topology change.