Inspired by recent developments in wave propagation and scattering experiments with parity-time (PT) symmetric materials, we discuss reciprocity and representation theorems for 3D inhomogeneous PT-symmetric materials and indicate some applications. We start with a unified matrix-vector wave equation which accounts for acoustic, quantum-mechanical, electromagnetic, elastodynamic, poroelastodynamic, piezoelectric and seismoelectric waves. Based on the symmetry properties of the operator matrix in this equation, we derive unified reciprocity theorems for wave fields in 3D arbitrary inhomogeneous media and 3D inhomogeneous media with PT-symmetry. These theorems form the basis for deriving unified wave field representations and relations between reflection and transmission responses in such media. Among the potential applications are interferometric Green’s matrix retrieval and Marchenko-type Green’s matrix retrieval in PT-symmetric materials.

Our research is in the tangent space of a point on a 4-dimensional Riemannian manifold. Besides the positive definite metric, the manifold is endowed with a tensor structure of type (1,1), whose fourth power is minus the identity. Both structures are compatible and they define an indefinite metric on the manifold. With the help of the indefinite metric we determine a circle in different 2-planes in the tangent space on the manifold, we also calculate the length and area of the circle. On a smooth closed curve such as a circle, we define a vector force field. Further, we obtain the circulation done by the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analogue of the well known Green’s formula in the Euclidean space.

The Marchenko method is a data-driven way which makes it possible to calculate Green's functions from virtual points in the subsurface by the reflection data at the surface, only requiring a macro velocity model. This method requires collocated sources and receivers. However, in practice, subsampling of sources or receivers will cause gaps and distortions in the obtained focusing functions and Green's functions. To solve this problem, this paper proposes to integrate sparse inversion into the iterative Marchenko scheme. Specifically, we add sparsity constraints to the Marchenko equations and apply sparse inversion during the iterative process. Our work not only reduces the strict requirements on acquisition geometries, but also avoids the complexity and instability of direct inversion for Marchenko equations. This new method is applied to a two-dimensional numerical example with irregular sampled data. The result shows that it can effectively fill gaps of the obtained focusing functions and Green's functions in the Marchenko method.

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

We propose a methodology for the quantification of model risk in the context of credit derivatives pricing and CVA, where the uncertain or unmodelled parameter is often the correlation between rates and credit. We take the rates model to be Hull-White (normal) and the credit model to be Black-Karasinski (lognormal). We show how highly accurate analytic pricing formulae, hitherto unpublished, can be derived for CDS and extended to address instruments with defaultable Libor flows which may in addition be capped and/or floored. We also consider the pricing of a contingent CDS with an interest rate swap underlying. We derive explicit expressions showing how to good accuracy the dependence of model prices on the uncertain parameter(s) can be captured in analytic formulae which are readily amenable to computation without recourse to Monte Carlo or lattice-based computation. In so doing, we take into account the impact on model calibration of the uncertain (or unmodelled) parameter.

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

In this work the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions we arrive at the following fourth-order differential equation

This paper presents an approach to the elastic analysis of beams subjected to Saint-Venant torsion using Green’s theorem and the finite difference method (FDM). The Saint-Venant torsion of beams, also called free torsion or unrestrained torsion, is characterized by the absence of axial stresses due to torsion; only shear stresses are developed. A solution to this torsion problem consists of finding a stress function that satisfies the governing equation and the boundary conditions. The FDM is an approximate method for solving problems described with differential equations; it does not involve solving differential equations, equations are formulated with values at selected nodes of the structure. In this paper, the beam’s cross-section was discretized using a two-dimensional grid and additional nodes were introduced on the boundaries. The introduction of additional nodes allowed us to apply the governing equations at boundary nodes and satisfy the boundary conditions. Beams with solid sections as well as multiply connected cross-sections were analyzed using this model; shear stresses and localized stresses at reentrant corners, torsion constant, and warping displacements were determined. Furthermore, beams with thin-walled closed sections, single-cell or multiple-cell, were analyzed using the Prandtl stress function whereby a linear distribution of the shear stresses over the thickness was considered; closed-form solutions for shear stresses and torsion constant were derived. The results obtained in this study showed good agreement with the exact results for rectangular cross-sections, and the accuracy was increased through a grid refinement. For thin-walled closed sections, the shear stresses obtained at the centerline using the closed-form solutions were in agreement with the values using Bredt’s analysis but the maximal values in the cross-section, which did not necessarily occur at the position with the smallest thickness, were higher; in addition, the results using the closed-form solutions were in good agreement with those using FDM.

In this review, we trace the evolution of the quantum spin-wave theory treating non-collinear spin configurations. Non-collinear spin configurations are consequences of the frustration created by competing interactions. They include simple chiral magnets due to competing nearest-neighbor (NN) and next-NN interactions and systems with geometry frustration such as the triangular antiferromagnet and the Kagomé lattice. We review here spin-wave results of such systems and also systems with the Dzyaloshinskii-Moriya interaction. Accent is put on these non-collinear ground states which have to be calculated before applying any spin-wave theory to determine the spectrum of the elementary excitations from the ground states. We mostly show results from a self-consistent Green’function theory to calculate the spin-wave spectrum and the layer magnetizations at finite T in two and three dimensions as well as in thin films with surface effects. Some new unpublished results are also included. Analytical details and the validity of the method are shown and discussed.