The new concept of an irrationality measure of sequences is introduced in this paper by means of the related irrational sequences. The main results are two criteria characterising lower bounds for the irrationality measures of certain sequences. Applications and several examples are included.

In this paper we consider a multivariate parallel profile model with polynomial growth curves. The covariance structure based on a random e¤ects model is assumed. The maximum likelihood estimators (MLE’s) are obtained under the random e¤ects covariance structure. The efficiency of the MLE is discussed.

We study reaction-di¤usion systems of propagator-controller type in the one-dimensional unit interval. When propagator di¤uses slowly, we establish the existence of transition layer equilibria by using singular perturbation expansions and a Lyapunov-Schmidt reduction method. Our approach to the existence also enables us to simultaneously obtain a stability criterion for the layer equilibria.

A singular perturbation problem for a reaction-di¤usion equation with a nonlocal term is treated. We derive an interface equation which describes the dynamics of internal layers in the intermediate time scale, i.e., in the time scale after the layers are generated and before the interfaces are governed by the volume-preserving mean curvature flow. The unique existence of solutions for the interface equation is demonstrated. A continuum of equilibria for the interface equation are identified and the stability of the equilibria is established. We rigorously prove that layer solutions of the nonlocal reaction-di¤usion equation converge to solutions of the interface equation on a finite time interval as the singular perturbation parameter tends to zero.

We introduce a 2-variable polynomial invariant for a virtual link derived from virtual magnetic graph diagrams, and using this invariant we prove the splitting of the Jones-Kau¤man polynomial with respect to the powers module four.

It is well known that the generalized linear mixed model is useful for analyzing the overdispersion and correlation structure for multivariate discrete data. In this paper, we derive an approximation of the density function for the generalized linear mixed model. This approximation is found to satisfy the properties of probability density function under some conditions. Therefore, this approximation can be regarded as a class of multivariate distributions. Estimation of the parameters in this class can be carried out by the maximum likelihood method. We give the likelihood ratio criteria for testing several covariance structures. Several simulation studies were also conducted for the Poisson log-normal model when the proposed density function is regarded as an approximate likelihood of the generalized linear mixed model.