This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique ref...

We derive formulas for (i) the number of distinct toroidal n × n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and (ii) the number of distinct toroidal n × n binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.

Combining the Mie scattering theory and a transfer matrix method, we investigate in detail the scattering of light by spherical Bragg "onion" resonators. We classify the resonator modes into two classes, the core modes that are confined by Bragg reflection, and the cladding modes that are confined by total internal reflection. We demonstrate that these two types of modes lead to significantly d...

We derive certain constraints on the reflection matrix for reflection from a plane, nonmagnetic, optically anisotropic surface using a reciprocity theorem stated long ago by Van de Hulst [Light Scattering by Small Particles (Wiley, 1957)] in the context of scattering of polarized light. The constraints are valid for absorbing and chiral media and can be used as tools to check the consistency of...

In this paper, we give the general crossing relation for boundary reflection matrix R(β) ,which is the extension of the work given by Ghoshal and Zamolodchikov .We also use the first non-trivial extended crossing relation to determine the scaler factor of R(β) which is the rational diagonal solution to the boundary Yang-Baxter equation in the case of l=2 and n=3.

We present a systematic way of constructing boundary quantum group generators associated to non-diagonal solutions of the reflection equation for the XXZ model. This is achieved by studying the asymptotics of the corresponding open transfer matrix. Linear intertwining relations for the K matrix and Sklyanin’s operator are also obtained, and the symmetry of the transfer matrix is investigated. L...

We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived.

A rigorous semi-analytical approach for specific microstructured optical fibers, which are formed by layered cylindrical arrays of circular rods symmetrically distributed on each of concentric circular rings, is presented for both of scalar-wave formulation and full-wave formulation. The method uses the T-matrix of a single circular rod, the reflection and transmission matrices for a cylindrica...