As a discrete-time quantum walk model on the one-dimensional integer lattice Z, the quantum walk recently constructed byWang and Ye [CaishiWang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2, by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z^d, which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

We study the spectral properties of Dirichlet Laplacian on the conical layer of the opening angle 2 and thickness equal to . We demonstrate that below the continuum threshold, which is equal to 1, there is an infinite sequence of isolated eigenvalues and analyse properties of these geometrically induced bound states. By numerical computation we find examples of the eigenfunctions.

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space-the natural extension of hyperbolic space by the de Sitter space-except for the degenerate case where all simplices are Euclidean in a generalized sense.

For a one-parameter family (V, \Omega_{i}) with I=1,...,p_g of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.

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