In this paper, a random graph process {$G$($t$)}_{$t$≥1}is studied and its degree sequence is analyzed. Let {$W$_{$t$}}_{$t$≥1}be an i.i.d. sequence. The graph process is defined so that, at each integer time$t$, a new vertex with$W$_{$t$}edges attached to it, is added to the graph. The new edges added at time$t$are then preferentially connected to older vertices, i.e., conditionally on$G$($t$-1), the probability that a given edge of vertex$t$is connected to vertex$i$is proportional to$d$_{$i$}($t$-1)+δ, where$d$_{$i$}($t$-1) is the degree of vertex$i$at time$t$-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_{W},τ_{P}}, where τ_{W}is the power-law exponent of the initial degrees {$W$_{$t$}}_{$t$≥1}and τ_{P}the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

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In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller-Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an mimes n rank mimes n matrix from mimes n number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant mimes n of the sensing operator is less than mimes n, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank mimes n matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

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