We consider the asymptotic behavior of a small solution to the linearized KdV equation. By rewriting this equation as a Hamiltonian system, the deduced Hamiltonian has unbounded, non-symmetric, and time-dependent potential. In this paper, we show the stableness of this solution to a linearized KdV equation in the $L^2$ sense and the decay estimates by analyzing this system.

Using a correspondence between the narrow ray class group modulo $m$ of a quadratic field and a certain set of equivalence classes of binary quadratic forms proved by Furuta and Kubota, we find a quadratic form $f$ and a pair of integers $(x_1,y_1)$ such that the norm of all integral ideals $\mathfrak{a}$ in a ray class is represented by $f(mx+x_1,my+y_1)$ with some integers $(x,y)$.

Many statisticians have considered various symmetry and asymmetry models to analyze square contingency tables with ordinal categories. We propose a generalized model, which indicates the asymmetric structure for cell probabilities. This model is the closest to the symmetry model in terms of the $f$-divergence under certain conditions. Then we decompose the symmetry model using the proposed model, and partition the likelihood-ratio chi-square test statistics for the symmetry model. The decomposition is useful to deduce the reason for a poor fit of the symmetry model.

In this paper, we consider $N(k)$-quasi Einstein manifolds satisfying the curvature conditions $\overline{P}(\xi ,X)\cdot W_2=0$ and $\overline{P}(\xi ,X)·H=0$ where $\overline{P}$, $W_2$ and $H$ are the pseudo projective, $W_2$ and conharmonic curvature tensors respectively. We study pseudo projectively symmetric $N(k)$-quasi Einstein manifolds and show that there does not exist a pseudo projectively semisymmetric $N(k)$-quasi Einstein manifold. Also, we construct some examples to support the existence of such manifolds.