Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.

Logic gates based on optical transistors

In this paper a simple and novel scheme to develop optical logic gates for fast switching. This can be obtained by sending two laser beams coaxially through nonlinear media with saturating nonlinearity. The nonlinearity in the optical properties on the medium is assumed to be cubic quintic nonlinearity. Semianalytical solutions of the wave equations for the fields of incident laser beams have been obtained by using a variational formalism. Emphasis are put on dynamical variations of beam widths of the laser beams with distance of propagation through the medium. It has been found that the weak beam modulates the intensity variations of the strong beam that giving rise to the possibility of an optical transistor. Combinations of these optical transistors can further be used to obtain logic gates.

A Multicriteria Motion Planning Approach for Combining Smoothness and Speed in Collaborative Assembly Systems

Human–robot interaction is an important aspect of Industry 4.0, and the extended use of robotics in industrial environments will not be possible without enabling them to safely interact with humans. This imposes relevant constraints in the qualitative characterization of the motions of robots when sharing their workspace with humans. In this paper, we address the trade-off between two such constraints, namely the smoothness, which is related to the cognitive stress that a person undergoes when interacting with a robot, and the speed, which is related to normative safety requirements. Given an execution time, such an approach will allow us to plan safe trajectories without neglecting cognitive ergonomics and production efficiency aspects. We first present the methodology able to express the balance between these qualities in the form of a composite objective function. Thanks to the variational formalism, we identify the related set of optimal trajectories with respect to the given criterion and give a suitable parametrization to them. Then, we are able to formulate the safety requirements in terms of a reparametrization of the motion. Finally, numerical and experimental results are provided. This allows the identification of the preferable sets of the possible motions that satisfy the operator’s psychological well-being and the assembly process performance by complying with the safety requirements in terms of mechanical risk prevention.

Quantum cosmology of a Hořava–Lifshitz model coupled to radiation

In the present paper, we canonically quantize a homogeneous and isotropic Hořava–Lifshitz cosmological model, with constant positive spatial sections and coupled to radiation. We consider the projectable version of that gravitational theory without the detailed balance condition. We use the Arnowitt–Deser–Misner (ADM) formalism to write the gravitational Hamiltonian of the model and the Schutz variational formalism to write the perfect fluid Hamiltonian. We find the Wheeler–DeWitt equation for the model, which depends on several parameters. We study the case in which parameter values are chosen so that the solutions to the Wheeler–DeWitt equation are bounded. Initially, we solve it using the Many Worlds interpretation. Using wave packets computed with the solutions to the Wheeler–DeWitt equation, we obtain the scalar factor expected value [Formula: see text]. We show that this quantity oscillates between finite maximum and minimum values and never vanishes. Such result indicates that the model is free from singularities at the quantum level. We reinforce this indication by showing that by subtracting one standard deviation unit from the expected value [Formula: see text], the latter remains positive. Then, we use the DeBroglie–Bohm interpretation. Initially, we compute the Bohm’s trajectories for the scale factor and show that they never vanish. Then, we show that each trajectory agrees with the corresponding [Formula: see text]. Finally, we compute the quantum potential, which helps understanding why the scale factor never vanishes.

Multi-fluid theory and cosmology: A convective variational approach to interacting dark-sector

The interaction of dark energy and dark matter has been studied widely using various formalisms in an effort to understand the physics of such gravitational interactions. Such studies are motivated by the idea that they might hold the key to resolving some of the outstanding problems in cosmology. We will consider the relativistic convective variational formalism in our study of dark matter (hereafter DM)-dark energy (hereafter DE) interaction. In particular, we go beyond the gravitational interaction and consider the potential entrainment phenomena involving the two dark-sector constituents. Ours is a formalism paper and focuses on the theoretical considerations that inform the modeling of such interactions.

ON THE USE OF EXTENDED PLATE THEORIES OF VEKUA – AMOSOV TYPE FOR WAVE DISPERSION PROBLEMS

he extended plate theory of I.N. Vekua – A.A. Amosov type is constructed on the background of the dimensional reduction approach and the Lagrangian variational formalism of analytical dynamics. The proposed theory allows one to obtain the hierarchy of refined plate models of different orders and to satisfy the boundary conditions on plates’ faces exactly by introducing the corresponding constraint equations into the Lagrangian model of two-dimensional continuum. The normal wave dispersion in an elastic layer is considered, the convergence of the two-dimensional solutions to the exact one is studied for the locking phase frequencies, the dimensionless stress distributions across the thickness of a layer are shown.

Propagation of Ultrashort Pulses in Nonlinear Media

In this paper, a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [9] has been used for the case of Kerr media. This equation which is called Generalized Nonlinear Schroedinger Equation usually has very complicated form and looking for its solutions is usually a very difficult task. Theoretical methods reviewed in this paper to solve this equation are effective only for some special cases. As an example we describe the method of developed elliptic Jacobi function expansion and its expended form: F-expansion Method. Several numerical methods of finding approximate solutions are briefly discussed. We concentrate mainly on the methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon, namely the collapse of solitons, where the variational formalism has been used.