abstract

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.

Open quantum walks (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quan- tum information and quantum computation. Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environ- ment of an open quantum system. In this paper, by using quantum Bernoulli noises as the environment, we introduce an open quantum walk on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an open quantum walk. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally we show links between the walk and a unitary quantum walk recently introduced in terms of quantum Bernoulli noises.

In this paper, we consider the numerical approximation of the three-dimensional poroelastic wave equations in the spherical coordinate system. One difficulty in the design of an efficient numerical scheme is that the problem is singular in the center and the polar axes of the computational domain. Nevertheless, we develop a hybrid finite difference/control volume method for solving this problem. Our method is explicit and is second order accurate in both space and time. Numerical results are shown to confirm the convergence rate of our method and the effectiveness to simulate wave propagation in poroelastic media in the spherical coordinate system.

We introduce a triangular array of integers defined by a(n, k) = k(k + 1)n−k for n ≥ 1 and 1 ≤ k ≤ n, which we call the researcher’s triangle. The construction is multiplicative and differs from classical additive triangles such as Pascal’s. We give combinatorial interpretations using a leaders-followers model and a rooted hierarchical structure model. We identify several diagonal sequences as classical figurate numbers, including powers of two (A000079), natural numbers (A000027), and oblong numbers (A002378), and establish connections to triangular and tetrahedral numbers. We derive a bivariate generating function and a recurrence relation, and we discuss the row sums.