Let $n > 1$ be an integer, $k > 1$ be an odd integer and $a > 0$ be an even integer. Suppose $a^{2} + b^{2}d = k^{n}$, where $d \neq 1, 3$ is a positive odd square-free integer and $\gcd(a, bd) = 1$. In this paper, we describe imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$ explicitly whose class numbers are divisible by $n$ if $d \equiv 1, 5, 7\bmod 8$ or $d \equiv 3\bmod 8$ with $(n, 3) = 1$ under certain conditions.

In this paper, we are concerned with the drift wave turbulence in a strong magnetic field. We prove the existence and uniqueness of a strong global in time solution to the initial boundary value problem for the model equations of drift wave turbulence similar to Hasegawa–Mima equation. Then we prove that the solution of Hasegawa–Wakatani equations established in [5] converges to that of Hasegawa–Mima like equation established at the first stage as the resistivity tends to zero on some time interval.

Let $L_{a}(x)$ be Lebesgue’s singular function with a real parameter $a$ ($0<a<1, a \neq 1/2$). As is well known, $L_{a}(x)$ is strictly increasing and has a derivative equal to zero almost everywhere. However, what sets of $x \in [0,1]$ actually have $L_{a}'(x)=0$ or $+\infty$? We give a partial characterization of these sets in terms of the binary expansion of $x$. As an application, we consider the differentiability of the composition of Takagi’s nowhere differentiable function and the inverse of Lebesgue’s singular function.

In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions. For the proof, we use a moduli of the valuative version of log mixed Hodge structures.

We present in this note an analogue of the Selberg-Weil-Kobayashi local rigidity Theorem in the setting of exponential Lie groups and substantiate two related conjectures. We also introduce the notion of stable discrete subgroups of a Lie group $G$ following the stability of an infinitesimal deformation introduced by T. Kobayashi and S. Nasrin (cf. [11]). For Heisenberg groups, stable discrete subgroups are either non-abelian or abelian and maximal. When $G$ is threadlike nilpotent, non-abelian discrete subgroups are stable. One major aftermath of the notion of stability as reveal some studied cases, is that the related deformation spaces are Hausdorff spaces and in most of the cases endowed with smooth manifold structures.

We solve a transversality problem relating to Bertelson-Gromov’s “dynamical Morse inequality”.