We estimate the misclassification probability of a Euclidean distance-based classifier in high-dimensional data. We discuss two types of estimator: a plug-in type estimator based on the normal approximation of misclassification probability (newly proposed), and an estimator based on the well-known leave-one-out cross-validation method. Both estimators perform consistently when the dimension exceeds the total sample size, and the underlying distribution need not be multivariate normality. We also numerically determine the mean squared errors (MSEs) of these estimators in finite sample applications of high-dimensional scenarios. The newly proposed plug-in type estimator gives smaller MSEs than the estimator based on leave-one-out cross-validation in simulation.

Let $G$ be an exponential solvable Lie group and $H$ a connected Lie subgroup of $G$. Given any discontinuous group $\mathit{\Gamma}$ for the homogeneous space $\mathscr M = G/H$ and any deformation of $\mathit{\Gamma}$, deformation of discrete subgroups may destroy proper discontinuity of the action on $\mathscr M$ as $H$ is not compact (except the case when it is trivial). To interpret this phenomenon in the case when $G$ is a 3-step nilpotent, we provide a layering of Kobayashi’s deformation space $\mathscr T(\mathit{\Gamma}, G, H)$ into Hausdorff spaces, which depends upon the dimensions of $G$-adjoint orbits of the corresponding parameter space. This allows us to establish a Hausdorffness theorem for $\mathscr T(\mathit{\Gamma}, G, H)$.

Let $G$ be a Lie-group and $\mathit{\Gamma} \subset G$ a cocompact lattice. For a finite-dimensional, not necessarily unitary representation $\omega$ of $\mathit{\Gamma}$ we show that the $G$-representation on $L^2(\mathit{\Gamma} \backslash G, \omega)$ admits a complete filtration with irreducible quotients. As a consequence, we show the trace formula for non-unitary twists and arbitrary locally compact groups.

We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coefficient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.

Masser and others have constructed sequences of ‘‘near miss’’ $abc$-triples, i.e., triples of relatively prime rational integers $(a, b, c)$ that asymptotically come close to violating the inequality that appears in the $abc$ Conjecture. In the present paper, we show various partial generalizations of Masser’s result to arbitrary number fields.

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points arising from three bitangent lines.