Shinichi Kobayashi found a generalization of the Coleman power series theory to formal groups of elliptic curves and applied it to a study of $p$-adic height pairings. In this paper, we generalize his theory of Coleman power series to general formal groups.

An $(I,J,K)$-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced by R. Bryant in order to separate and clarify the local and global aspects in Finsler geometry making use of Cartan's method of exterior differential systems. In this paper, we show that there is a close relation between $(I,J,1)$-generalized Finsler structures and a class of contact circles, namely the so-called Cartan structures.This correspondence allows us to determine the topology of 3-manifolds that admit $(I,J,1)$-generalized Finsler structures and to single out classes of $(I,J,1)$-generalized Finsler structures induced by standard Cartan structures.

We construct a family of McKay quiver representations on the Danilov resolution of the $\frac{1}{r}(1,a,r-a)$ singularity. This allows us to show that the resolution is the normalization of the coherent component of the fine moduli space of $\theta$-stable McKay quiver representations for a suitable stability condition $\theta$. We describe explicitly the corresponding union of chambers of stability conditions for any coprime numbers $r,a$.

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.

For an algebroid function in the unit disk of finite lower order with a deficient value, we can estimate its growth order in terms of the convergence exponent of the points of the deficient value and other distinct values not lying on a radial system and the maximal difference of the arguments of adjacent rays.

In 1984 Masaru Takeuchi showed that every real form of a hermitian symmetric space of compact type is a symmetric $R$-space and vice-versa. In this note we present a geometric proof of this result.

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted (=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore, by using the Cartan type identity, we show that certain kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of non-compact type are principal orbits of Hermann actions.

The convergence of stochastic processes is one of subjects founded on importance of the numerical analysis and physical models with stability. Such practical importance inspires us with vast range of interests as to on which space the convergence can be addressed and which sort of accommodated method is required for demonstrating the convergence on the space in the focus. In this article, we establish an accommodated procedure to show the convergence of Markov processes on the ring of $p$-adic integers which emerges from a construction of random fractals. As seen in other studies on the subject, the notion of generalized Mosco-convergence will be highlighted.