We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log$n$, with$n$being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the$negative$solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view of its possible extensions to other kinds of collective motion.

Processing streaming data as they arrive is often necessary for high dimensional data analysis. In this paper, we analyze the convergence of a subspace online PCA iteration, as a followup of the recent work of Li, Wang, Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who considered the case for the most significant principal component only, i.e., a single vector. Under the sub-Gaussian assumption, we obtain a finite-sample error bound that closely matches the minimax information lower bound of Vu and Lei [Ann. Statist. 41:6 (2013), 2905-2947].

There is an unmet medical need to identify neuroimaging biomarkers that allow us to accurately diagnose and monitor Alzheimer's disease (AD) at its very early stages and to assess the response to AD-modifying therapies. To a certain extent, volumetric and functional magnetic resonance imaging (fMRI) studies can detect changes in structure, cerebral blood flow, and blood oxygenation that distinguish AD and mild cognitive impairment (MCI) subjects from healthy control (HC) subjects. However, it has been challenging to use fully automated MRI analytic methods to identify potential AD neuroimaging biomarkers. We have thus proposed a method based on independent component analysis (ICA) for studying potential AD-related MR image features that can be coupled with the use of support vector machine (SVM) for classifying scans into categories of AD, MCI, and HC subjects. The MRI data were selected from

We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori’s Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj [20] for automorphisms on 3-dimensional projective manifolds, and also determine π1(X).