High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.

The global existence of solutions in H 2 is well known for H 2 critical nonlinear Schrödinger equations with small initial data in high dimensions d ≥ 8(see [4]). However, even though the solution is constructed by a fixed-point technique, continuous dependence in H 2 does not follow from the contraction mapping argument. Comparing with the low dimension cases 4 < d < 8, there is an obstruction to this approach because of the sub-quadratic nature of the nonlinearity(which makes the derivative of the nonlinearity non-Lipschitz). In this paper, we resolve this difficulty by applying exotic Strichartz spaces of lower order instead and show that the solution depends continuously on the initial value in the sense that the local flow is continuous H 2 → H 2 .

In this paper, we consider the Neumann boundary value problem for a system of two elliptic equations involving the critical Sobolev exponents. By means of blowing-up method, we obtain behavior of positives with low energy and asymptotic behavior of positive solutions with minimum energy as the parameters λ, µ → ∞.

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In this paper, Fourier spectral method combined with modified fourth order exponential time-differencing Runge-Kutta is proposed to solve the nonlinear Schrödinger equation with a source term. The Fourier spectral method is applied to approximate the spatial direction, and fourth order exponential time-differencing Runge-Kutta method is used to discrete temporal direction. The proof of the conservation law of the mass and the energy for the semidiscrete and full-discrete Fourier spectral scheme is given. The error of the semidiscrete Fourier spectral scheme is analyzed in the proper Sobolev space. Finally, several numerical examples are presented to support our analysis.