Essential spectra of operator matrices and applications

The main goal of this paper is to give a characterization of Wolf, Schechter, and Browder essential spectra of (2 × 2) matrix operator on a Banach space. Furthermore, we apply the obtained results to transport equations. Copyright © 2013 John Wiley & Sons, Ltd.     

The Magnetocaloric Effect and Heat Capacity of Suspensions of High-Dispersity Samarium Ferrite

The magnetocaloric effect and specific heat capacity of an aqueous suspension of samarium ferrite were determined calorimetrically over the temperature range 288–343 K in magnetic fields of 0–0.7 T. The data obtained were used to calculate changes in the magnetic component of the molar heat capacity and entropy of the magnetic phase and changes in the enthalpy of the process under an applied magnetic field. The magnetocaloric effect was found to increase nonlinearly as the magnetic field induction grew. The corresponding temperature dependences contained a maximum at 313 K related to the second-order magnetic phase transition at the Curie point. The field and temperature dependences of heat capacity contained a maximum in fields of 0.4 T and a minimum at the magnetic phase transition temperature.     

Holographic instant conformal symmetry breaking by colliding conical defects

We study instant conformal symmetry breaking as a holographic effect of ultrarelativistic particles moving in the AdS₃ space–time. We give a qualitative picture of this effect based on calculating the two-point correlation functions and the entanglement entropy of the corresponding boundary theory. We show that in the geodesic approximation, because of gravitational lensing of the geodesics, the ultrarelativistic massless defect produces a zone structure for correlators with broken conformal invariance. At the same time, the holographic entanglement entropy also exhibits a transition to nonconformal behavior. Two colliding massless defects produce a more diverse zone structure for correlators and the entanglement entropy.     

Measure of weak noncompactness, some new properties in Fredholm theory, characterization of the Schechter essential spectrum and application to transport operators

Measures of weak noncompactness have been successfully applied in topology, functional analysis and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We work with the notion of the measures of weak noncompactness in order to establish some results concerning the class of semi-Fredholm and Fredholm operators. Further, we apply the obtained results to prove, under some conditions on the perturbed operator, the invariance of the Schechter essential spectrum on Banach spaces. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.     

Asymptotic expansion of solutions in a rolling problem

Asymptotic methods in the theory of differential equations and in nonlinear mechanics are commonly used to improve perturbation theory in the small oscillation regime. However, in some problems of nonlinear dynamics, in particular for the Higgs equation in field theory, it is important to consider not only small oscillations but also the rolling regime. In this article we consider the Higgs equation and develop a hyperbolic analogue of the averaging method. We represent the solution in terms of elliptic functions and, using an expansion in hyperbolic functions, construct an approximate solution in the rolling regime. An estimate of accuracy of the asymptotic expansion in an arbitrary order is presented.     

A characterization of some subsets of Schechter's essential spectrum and application to singular transport equation

The purpose of this paper is to provide a detailed treatment of some subsets of Schechter's essential spectrum of closed, densely defined linear operators subjected to additive perturbations. Our results are used to describe the essential approximate point spectrum and the essential defect spectrum of singular neutron transport operators in bounded geometries.