A subgroup H of a group G is said to be pronormal in G if each of its conjugates $H^g$ in G is already conjugate to it in the subgroup $⟨H,H^g⟩$. Extending the well-known class of metahamiltonian groups, we study soluble groups in which every subgroup is abelian or pronormal.

We present a new approach for the proof of some formulas of Takuro Shintani concerning the Kronecker limit formulas for real quadratic fields.

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural limiting mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotent orbit in several variables.

By using new techniques with the degenerate Whittaker functions developed by Ikeda and Yamana, we construct a higher-level cusp form on ${\mathrm{GE}}_{7,3}$ (exceptional similitude group of type $E_7$), called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes our earlier results on level-1 forms. But there is a new phenomenon in higher levels; we can handle cusp forms with nontrivial central characters. We also compute the degree-133 adjoint L-function of the Ikeda type lift.

We prove that any symplectic automorphism of finite order on a manifold of type $\mathrm{OG}6$ acts trivially on the Beauville–Bogomolov–Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices.

In this paper, we study a discrete approximation to Brownian motion with varying dimension (BMVD) introduced by Chen and Lou in their 2019 paper with continuous time random walks on square lattices. The state space of BMVD contains a 2-dimensional component, a 3-dimensional component, and a “darning point” which joins these two components. Such a state space is equipped with the geodesic distance under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous-time reversible random walks with exponential holding times.

The Chebyshev polynomials $T_d$ in one variable are typical chaotic maps on $\mathbb{C}$. Chebyshev endomorphisms $P_{A_n}^d:{\mathbb{C}}^n\to {\mathbb{C}}^n$ are also chaotic. We consider the action of the dihedral group $D_{n+1}$ on ${\mathbb{C}}^n$. The endomorphism $P_{A_n}^d$ maps any $D_{n+1}$-orbit of $\mathbf{z}\in {\mathbb{C}}^n$ to a $D_{n+1}$-orbit of $P_{A_n}^d(\mathbf{z})$. The endomorphism $P_{A_n}^d$ induces a mapping on ${\mathbb{C}}^n∕D_{n+1}$.

Using invariant theory, we embed ${\mathbb{C}}^n∕D_{n+1}$ as an affine subvariety X in ${\mathbb{C}}^m$. Then we have morphisms $g_d$ on X. We study the cases $n=2$ and 3. In these cases, the morphisms $g_d$ are defined over $\mathbb{Z}$. We find a class of affine subvarieties V of X which are invariant under $g_d$. These varieties are concerned with branch loci or critical loci. The class contains ${\mathbb{C}}^2$, a cuspidal cubic, a parabola, a quadric hypersurface in ${\mathbb{C}}^4$, an affine algebraic surface in ${\mathbb{C}}^4$ which is birationally equivalent to an affine quadric cone in ${\mathbb{C}}^3$, and others. For each affine variety V in the class, there exists a polynomial parameterization $P_V$ satisfying $g_d{|}_V(P_V(y_1,\dots ,y_k))=P_V(T_d(y_1),\dots ,T_d(y_k))$, where $T_d(z)$ is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of $g_d{|}_V$ in each invariant set V and give relations between them.