Let $C_m$ be a cyclic group of order $m$. We prove that if a group $G$ fits into an extension $1 \rightarrow C^2_{2^{n+1}} \rightarrow G \rightarrow C_2 \rightarrow 1$ for $n\geq 1$ then $G$ is good in the sense of Hopkins-Kuhn-Ravenel, i.e., $K(s)^\ast(BG)$ is evenly generated by transfers of Euler classes of complex representations of subgroups of $G$.

In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N \times \mathbb R$, submanifolds with bounded $\varpi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p$-Laplacian.

We calculate the twisted Alexander polynomials of $(−2, 3, 2n+1)$-pretzel knots associated to the family of their $SL_2(\mathbb C)$-representations which contains their holonomy representations.

Let $(M,g_M, \mathscr F)$ be a closed, connected Riemannian manifold with a Riemannian foliation $\mathscr F$ of nonzero constant transversal scalar curvature. When $M$ admits a transversal nonisometric conformal field, we find some generalized conditions that $\mathscr F$ is transversally isometric to $(S^q(1/c),G)$, where $G$ is the discrete subgroup of $O(q)$ acting by isometries on the last $q$ coordinates of the sphere $S^q(1/c)$ of radius $1/c$.

We prove the analytic smoothing e¤ect for solutions to the system of nonlinear Schrödinger equations under the gauge invariant nonlinearities. This result extends the known result due to Hoshino [Nonlinear Differential Equations Appl. 24 (2017), Art. 62]. Under rapidly decaying condition on the initial data, the solution shows a smoothing effect and is real analytic with respect to the space variable. Our theorem covers not only the case for the gauge invariant setting but also multiple component case with higher power nonlinearity up to the fifth order.

In this paper, we consider a model selection criterion using the GEE method including unknown scale and correlation parameters. We propose a model selection criterion for selecting variables and a working correlation structure. Under some regularity conditions, we showed that our criterion is the same as the criterion proposed by Inatsu and Imori [8]. A numerical study reveals that we can reduce the prediction error by selecting both variables and a working correlation structure.

In this article, we introduce the space of formal power series of class $\mathbb M$ of Roumieu (resp., Beurling) type, which is the generalization of the space of entire functions of normal (resp., minimal) type with respect to a proximate order. And we characterize continuous endomorphisms of these spaces.