In [8] we classified all ``convex orders'' on the positive root system $\Delta_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and gave a concrete method of constructing all convex orders on $\Delta_+$. The aim of this paper is to give a new description of ``convex bases'' of PBW type of the positive subalgebra $U^+$ of the quantum affine algebra $U=U_q({\mathfrak g})$ by using the concrete method of constructing all convex orders on $\Delta_+$. Applying convexity properties of the convex bases of $U^+$, for each convex order on $\Delta_+$, we construct a pair of dual bases of $U^+$ and the negative subalgebra $U^-$ with respect to a $q$-analogue of the Killing form, and then present the multiplicative formula for the universal $R$-matrix of $U$.

We study periodic travelling wave solutions of a derivative non-linear Schrödinger equation and show the existence of infinitely many families of semi-trivial solutions (Theorem 2). Each of the families constitutes a branch of travelling waves corresponding to a non-zero integer called the winding number. A sufficient condition for the orbital stability of travelling waves on the branches with positive winding number is given in terms of the wave speed and winding number of the solution (Theorem 3). Bifurcation points are found on each semi-trivial branch of travelling wave solutions (Theorem 4), and the qualitative, and approximately quantitative, orbitshapes of the bifurcated solutions are given. The stability of the semi-trivial solutions under subharmonic perturbations is studied in Theorem 6, and subharmonic bifurcations are established in Theorem 7.

For the normalized generalized matrix function $\overline d_{\chi}^{G} (A)$ for $3 \times 3$ positive semi-definite Hermitian matrices $A$, the permanental dominance conjecture $\per A \geq \overline d_{\chi}^{G} (A)$ is known to hold. In this paper, we show that this inequality is not sharp, and give a sharper bound.

This paper is concerned with discovering linear functional relationships among $k$ $p$-variate populations with mean vectors $\vmu_{i}$, $i=1,\ldots ,k$ and a common covariance matrix $\Sigma$. We consider a linear functional relationship to be one in which each of the specified $r$ mean vectors, for example, $\vmu_{1}, \ldots, \vmu_{r}$ are expressed as linear functions of the remainder mean vectors $\vmu_{r+1}, \ldots, \vmu_{k}$. This definition differs from the classical linear functional relationship, originally studied by Anderson [1], Fujikoshi [8] and others, in that there are $r$ linear relationships among $k$ mean vectors without any specification of $k$ populations. To derive our linear functional relationship, we first obtain a likelihood test statistic when the covariance matrix $\Sigma$ is known. Second, the asymptotic distribution of the test statistic is studied in a high-dimensional framework. Its accuracy is examined by simulation.

We give useful criteria for lips, beaks and swallowtail singularities of a smooth map from the plane into the plane. As an application of the criteria, we discuss the singularities for a Cauchy problem of single conservation law.

The Dunkl transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price’s theorem, Beurling’s theorem and Donoho-Stark’s uncertainty principle are obtained for the Dunkl transform.