Abstract:

I will consider solutions u_ of the three-dimensional Navier–Stokes equations on the periodic domains Q_ := (, )^3 as the domain size , and compare them to solutions of the same equations on the whole space. For compactly-supported initial data u_^0 H^1 (Q_), an appropriate extension of u_ converges to a solution u of the equations on R^3, strongly in L^r(0, T; H^1(R^3)), r [1, 4) (the result is in fact more general than this). The same also holds when u_^0 is the velocity corresponding to a fixed, compactly-supported vorticity. Such convergence is sufficient to show that if an initial compactly-supported velocity u_0 H^1(R^3) or an initial compactly-supported vorticity _0 H^1(R^3) gives rise to a smooth solution on [0, T] for the equations posed on R^3, a smooth solution will also exist on [0, T] for the same initial data for the periodic problem posed on Q^ for sufficiently large; this illustrates a ‘transfer of regularity’ from the whole space to the periodic case.

Abstract:

Most existing statistical network analysis literature assumes a global view of the network, under which community detection, testing, and other statistical procedures are developed. Yet in the real world, people frequently make decisions based on their partial understanding of the network information. As individuals barely know beyond friends’ friends, we assume that an individual of interest knows all paths of length up to L = 2 that originate from her. As a result, this individual’s perceived adjacency matrix B differs significantly from the usual adjacency matrix A based on the global information. The new individual-centered partial information framework sparks an array of interesting endeavors from theory to practice. Key general properties on the eigenvalues and eigenvectors of BE, a major term of B, are derived. These general results, coupled with the classic stochastic block model, lead to a new theory-backed spectral approach to detecting the community memberships based on an anchored individual’s partial information. Real data analysis delivers interesting insights that cannot be obtained from global network analysis. (joint with Xiao Han)