In addition to the conventional renormalized-coupling-constant picture, point interactions in two and three dimensions are shown to model within a suitable energy range scattering on localized potentials, both attractive and repulsive.

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.

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.

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In this paper, we first present a theoretical analysis model on the Proof-of-Work (PoW) for cryptocurrency blockchain. Based on this analysis, we present a new type of PoW, which relies on the hardness of solving a set of random quadratic equations over the finite field GF(2). We will present the advantages of such a PoW, in particular, in terms of its impact on decentralization and the incentives involved, and therefore demonstrate that this is a new good alternative as a new type for PoW in blockchain applications.