Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a (simplicial, polyhedral or cellular) complex without closed orbits, where each cell may either have no arrows, receive a single arrow from one of its facets, or conversely, send a single arrow into a cell of which it is a facet. By following arrows, one can then construct a natural Floer-type boundary operator. Here, we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows (but not both), again in the absence of closed orbits. Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.

Joint parsimonious modeling the mean and covariance is important for analyzing longitudinal data, because it accounts for the efficiency of parameter estimation and easy interpretation of variability. The main potential risk is that it may lead to inefficient or biased estimators of parameters while misspecification occurs. A good alternative is the semiparametric model. In this paper, a Bayesian approach is proposed for modeling the mean and covariance simultaneously by using semiparametric models and the modified Cholesky decomposition. We use a generalized prior to avoid the knots selection while using B-spline to approximate the nonlinear part and propose a Markov Chain Monte Carlo scheme based on Metropolis–Hastings algorithm for computations. Simulation studies and real data analysis show that the proposed approach yields highly efficient estimators for the parameters and nonparametric parts in the mean, meanwhile providing parsimonious estimation for the covariance structure.

The study of real-life network modeling has become very popular in recent years. An attractive model is the scale-free percolation model on the lattice ${\mathbb Z}^d$, $d\ge 1$, because it fulfills several stylized facts observed in large real-life networks. We adopt this model to continuum space which leads to a heterogeneous random-connection model on ${\mathbb R}^d$: Particles are generated by a homogeneous marked Poisson point process on ${\mathbb R}^d$, and the probability of an edge between two particles is determined by their marks and their distance. In this model we study several properties such as the degree distributions, percolation properties and graph distances.

We consider estimation of the scale parameter of a two-parameter exponential distribution on the basis of doubly censored data. Classes of estimators, improving upon the minimum risk equivariant estimator, are derived under an arbitrary strictly convex loss function. Some existing dominating procedures are shown to belong to the proposed classes of estimators.

In this paper, we consider the Neumann problem for special Lagrangian equations with critical phase. The global gradient and Hessian estimates are obtained. Using the method of continuity, we prove the existence of solutions to this problem.

A new method to design a cubic Pythagorean-hodograph (PH) spline curve from any given control polygon is proposed. The key idea is to suitably choose a set of auxiliary points associated with the edges of the given control polygon to guarantee the constructed PH spline has $G^1$ continuity or curvature continuity. The method facilitates intuitive and efficient construction of open and closed cubic PH spline curves that typically agrees closely with the same friendly interface and properties as B-splines, for example, the convex hull and variation-diminishing properties.