We survey recent progress on the case of the Cauchy problem for the generalized reduced Ostrovsky equation $u_t=S({\partial }_x)u+{(f(u))}_x$, where the operator $S({\partial }_x)$ is defined through the Fourier transform as $S({\partial }_x)={ℱ}^{-1}\frac{1}{i\xi }ℱ$, and the nonlinear interaction is given by $f(u)=|u{|}^{\rho -1}u$ if is not an integer and $f(u)=u^{\rho }$ if is an integer.

This is a survey for the research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry.

For analyzing square contingency tables, Bowker [14] proposed the symmetry model. Caussinus [16] proposed the quasi-symmetry model and gave a decomposition of model such that the symmetry model holds if and only if both the quasi-symmetry and the marginal homogeneity models hold. Bhapkar and Darroch [13] gave the similar theorem for multi-way contingency tables. For square tables and for multi-way tables, the present paper (1) reviews various models of symmetry and asymmetry, (2) reviews the decompositions of models, (3) gives some figures which indicate the relationships among various models, and (4) gives a new decomposition of symmetry model.

In this decade, nonlinear conjugate gradient methods have been focused on as effective numerical methods for solving large-scale unconstrained optimization problems. Especially, nonlinear conjugate gradient methods with the sufficient descent property have been studied by many researchers. In this paper, we review sufficient descent nonlinear conjugate gradient methods.

SIRS type epidemiological model has a fundamental form to study the role of temporal immunity of recovered individuals in disease transmission dynamics and several variant models have been considered in the last century, but up to now dynamical aspects of the model are not fully elucidated. We here look over previous studies concerning qualitative analysis for a family of SIRS type epidemiological models. To this aim we construct a general model in a form of delay equation, a coupled system of a renewal equation and delay differential equations, structuring infected population by infection-age (time elapsed since infection). We re-examine the structure of equilibria and stability of the disease free equilibrium. We then introduce slightly improved stability conditions for an endemic equilibrium. Specifying modelling ingredients we derive two special cases where the model can be represented as a system of ordinary and delay differential equations that have appeared in the literature. For those models we have a powerful tool, namely Lyapunov function, to study global stability of the endemic equilibrium. Relating epidemic models are also discussed.

We discuss multivariate multiple comparison procedure among mean vectors. In general, it is difficult to find the exact critical value that is used for this problem. So, some approximation procedures have been discussed by many authors. In this paper, we review some results concerning the following approximation procedures: (i) multivariate Tukey-Kramer type procedure and (ii) approximation procedure based on Bonferroni’s inequality.

In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real analytic pseudo-Riemannian manifold. In particular, for a pseudo-Riemannian homogeneous space, we define another complexification and a (complete) anti-Kaehler metric on the complexification. One of main purposes of this paper is to find the interesting relation between these two complexifications (equipped with the anti-Kaehler metrics) of a pseudo-Riemannian homogeneous space. Another of main purposes of this paper is to show that almost all principal orbits of some isometric action on the first complexification (equipped with the anti-Kaehler metric) of a semi-simple pseudo-Riemannian symmetric space are curvature-adapted isoparametric submanifolds with flat section in the sense of this paper.

For square contingency tables with nominal categories, Tomizawa (1994), and Tomizawa, Seo and Yamamoto (1998) considered the power-divergence-type measures to represent the degree of departure from symmetry. The present paper proposes the Wald-type measure to represent the degree of departure from symmetry. In sample version the proposed measure is obtained by modifying the Wald test statistic by the upper limit with fixed sample size. The paper also describes the relationship between the proposed measure and asymmetry models, and the relationship between the measure and the bivariate normal distribution. Examples are given.

We propose a new approach for constructing a survival tree which is used to analysis of time to event data. The construction of survival tree is performed by the classification and regression tree (CART) algorithm, which constructs a tree model by recursively dividing the data. In the traditional approach, internal-node data are divided by a splitting rule which consider only one covariate for each node. In the proposed approach, two or more combinations of covariates are considered for dividing the internal-node data in the algorithm. We show that the proposed method is more suitable than the traditional approach in some situations through the comparative research by simulations. We also present the result of an actual analysis based on proposed approach.

We study affine immersions with twistor lifts. Using a decomposition of a connection, we obtain several projective invariants for such affine immersions. In particular, affine immersions with holomorphic twistor lifts are considered. We can show the property that an affine immersion has holomorphic twistor lifts is invariant under projective transformations and characterize immersions with holomorphic twistor lifts by vanishing of some of projective invariants. In the case of compact affine surfaces with holomorphic twistor lifts, we see a quantization phenomenon for one of the projective invariants which we obtain. Moreover, we prove that a real analytic twistor holomorphic affine surface with the symmetric Ricci tensor with respect to both complex structures is totally geodesic or totally umbilic.

We construct a nested row-column design with split units for a two-factor experiment. We use a balanced incomplete block design with nested rows and columns (BIBRC for short) repeatedly for the whole plot treatments and we use a proper block design for the subplot treatments. We give the stratum efficiency factors for such a nested row-column design with split units, which has the general balance property.

We study harmonicity and pluriharmonicity of holomorphic maps in almost contact geometry.

In this article, we determine the wave front sets of solutions to time dependent Schrödinger equations with a sub-quadratic potential by using the representation of the Schrödinger evolution operator via wave packet transform (short time Fourier transform).

We study the orbital stablity and instability of solitary wave solutions for nonlinear Schrödinger equations of derivative type.

A graph is said to be Hamiltonian-connected if there exists a Hamiltonian path between any given pair of distinct vertices. In 1983, Thomassen proved that every 4-connected plane graph is Hamiltonian-connected, using the concept of Tutte subgraph. In this paper, we give a new proof to Thomassen’s theorem.

In this paper we give a lower bound of the circumference of a graph in terms of girth and the number of edges. It is shown that a graph of girth at least $g\ge 4$ with $n$ vertices and at least $m\ge n$ edges contains a cycle of length at least $(g-2)m/(n-2)$. In particular, for the case $g=4$, an analogous result for 2-edge-connected weighted graphs is given. Moreover, the extremal case is characterized in both results.

In this paper we describe the Hochschild cohomology rings for algebras in a class of some special biserial algebras which contains a cluster-tilted algebra of Dynkin type ${\mathbb{D}}_4$. In particular it is shown that the Hochschild cohomology rings modulo nilpotence for these algebras are isomorphic to the polynomial ring $K[x]$. As an application we prove that the cluster-tilted algebra of type ${\mathbb{D}}_4$ contained in this class satisfies the finiteness conditions (Fg1) and (Fg2) introduced in [EHSST].

This paper provides the approximations for the tail quantile for hybrid lognormal distribution which is often applied to the distribution for some datasets such as the radiation dose. We derive the Mills ratio for the hybrid lognormal distribution, and have the main results on the basis of the Mills ratio using the idea stated in [7]. Finally, the numerical evaluation for the main results is given.

Skewness and kurtosis characteristics of a multivariate $p$-dimensional distribution introduced by Mardia (1970) have been used in various testing procedures and demonstrated attractive asymptotic properties in large sample settings. However these characteristics are not designed for high-dimensional problems where the dimensionality, $p$ can largely exceeds the sample size, $N$. Such type of high-dimensional data are commonly encountered in modern statistical applications. This the suggests that new measures of skewness and kurtosis that can accommodate high-dimensional settings must be derived and carefully studied. In this paper, we show that, by exploiting the dependence structure, new expressions for skewness and kurtosis are introduced as an extension of the corresponding Mardia’s measures, which uses the potential advantages that the block-diagonal covariance structure has to offer in high dimensions. Asymptotic properties of newly derived measures are investigated and the cumulant based characterizations are presented along with of applications to a mixture of multivariate normal distributions and multivariate Laplace distribution, for which the explicit expressions of skewness and kurtosis are obtained. Test statistics based on the new measures of skewness and kurtosis are proposed for testing a distribution shape, and their limit distributions are established in the asymptotic framework where $N\to \infty$ and $p$ is fixed but large, including . For the dependence structure learning, the gLasso based technique is explored followed by AIC step which we propose for optimization of the gLasso candidate model. Performance accuracy of the test procedures based on our estimators of skewness and kurtosis are evaluated using Monte Carlo simulations and the validity of the suggested approach is shown for a number of cases when .