We provide an irreducibility criterion for polynomials in one variable over a valued field $K$, in the presence of a secondary valuation defined on $K$.

In this paper some continuation techniques based on the implicit function theorem are combined with the topological degree to show the existence of solution isolas, with respect to a given state, for a class of weighted boundary value problems of superlinear indefinite elliptic type. No result of this nature seems to be available in the literature. Further, pseudo-spectral methods coupled with path following solvers are used to compute these isolas in some simple one-dimensional prototype models.

For the classification problem between two normal populations with a common covariance matrix, we consider a class of discriminant rules based on a general discriminant function $T$. The class includes the one based on Fisher’s linear discriminant function and the likelihood ratio rule. Our main purpose is to derive an optimal discriminant rule by using an asymptotic expansion of misclassification probability when both the dimension and the sample sizes are large. We also derive an asymptotically unbiased estimator of the misclassification probability of $T$ in our class.