Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-\'Emery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(\Delta_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^\alpha(x,t) = 0, \end{align*} where $(x,t) \in M^N \times (-\infty, \infty)$ and $\alpha$ is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions.Comment: 18 page

A generalized Gaussian process model for computer experiments with binary time series

Non-Gaussian observations such as binary responses are common in some computer experiments. Motivated by the analysis of a class of cell adhesion experiments, we introduce a generalized Gaussian process model for binary responses, which shares some common features with standard GP models. In addition, the proposed model incorporates a flexible mean function that can capture different types of time series structures. Asymptotic properties of the estimators are derived, and an optimal predictor as well as its predictive distribution are constructed. Their performance is examined via two simulation studies. The methodology is applied to study computer simulations for cell adhesion experiments. The fitted model reveals important biological information in repeated cell bindings, which is not directly observable in lab experiments.Comment: 49 pages, 4 figure

Bis(2,2′-bipyridine)-1κ2 N,N′;3κ2 N,N′-hexa-μ-methacrylato-1:2κ6 O:O′;2:3κ6 O:O′-(nitrato-2κ2 O,O′)-1,3-dicobalt(II)-2-terbium(III)

In the title trinuclear cobalt–terbium complex, [Co2Tb(C4H5O2)6(NO3)(C10H8N2)2], the central TbIII and each of the CoII ions are bridged by three carboxyl­ate groups of the methacrylate anions. The TbIII cation is coordinated by six O atoms from six methacrylate anions and two O atoms from a chelating nitrate anion in a distorted square-anti­prismatic geometry. Each CoII ion is coordinated by three O atoms from three methyl­acrylate anions and two N atoms of a 2,2′-bypiridine ligand in a distorted square-pyramidal geometry. In the crystal structure, π–π stacking between the pyridine rings [centroid–centroid distances = 3.682 (8) and 3.760 (8) Å] is observed and weak inter­molecular C—H⋯O hydrogen bonding is also present