The main purpose of this paper is to determine the coarse orbit types of all non-Kählerian, symplectic homogeneous spaces $G/H$ with $G$ non-compact simple and $H$ compact. Our result, together with the results of Wang and Yichao, enables one to know all dual manifolds of every Kähler $C$-space.

In this paper, we are concerned with oscillation of solutions of a certain class of third-order nonlinear delay differential equations of the form $% x^{^{\prime \prime \prime }}(t)+p(t)x^{^{\prime}}(t)+q(t)f(x(\tau (t)))=0$. We establish some new oscillation results that extend and improve some results in the literature in the sense that our results do not require that $% \tau ^{^{\prime }}(t)\geq 0$. Some examples are considered to illustrate the main results and some conjectures and open problems are presented.

In the present study, we consider the selection of model selection criteria for multivariate ridge regression. There are several model selection criteria for selecting the ridge parameter in multivariate ridge regression, e.g., the $C_p$ criterion and the modified $C_p$ ($MC_p$) criterion. We propose the generalized $C_p$ ($GC_p$) criterion, which includes $C_p$ and $MC_p$ criteria as special cases. The $GC_p$ criterion is specified by a non-negative parameter $\lambda$, which is referred to as the penalty parameter. We attempt to select an optimal penalty parameter such that the predicted mean square error (PMSE) of the predictor of ridge regression after optimizing the ridge parameter is minimized. Through numerical experiments, we verify that the proposed optimization methods exhibit better performance than conventional optimization methods, i.e., optimizing only the ridge parameter by minimizing the $C_p$ or $MC_p$ criterion.

Using an integral formula on a homogeneous Siegel domain, we give a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact in terms of a boundary behavior of the Bergman kernel.

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.