In this paper, we first investigate the shape operators of certain kind of orbits of the isotropy action of a semi-simple pseudo-Riemannian symmetric space. The investigation is performed by investigating the complexified action. Next, by using the fact obtained by the investigation, we show that certain kind of principal orbits of a Hermann type action on a semi-simple pseudo-Riemannian symmetric space are curvature-adapted proper complex equifocal submanifolds and that their shape operators are semisimple. It follows from this fact that the principal orbits are isoparametric submanifolds with flat section. Also, we derive an interesting structure of a semi-simple pesudo-Riemannian symmetric space (in particular, the complexification of a Riemannian symmetric space) from two special Hermann type actions on the space.

In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined in [4] and [2]. One is the relatively exact 2-category defined in [4], and the other(s) is the (2-)abelian Gpd-category defined by Dupont [2]. We compare these notions, using the arguments in [4] and [2]. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories.

In this paper, we determine the Cartier dual of certain finite group schemes of type (p ⁿ , ..., p ⁿ ), restricting ourselves to positive characteristic p case. They are given by the kernel of certain endomormphisms of the fibre product W _l,A ×_{Spec A} … ×_{Spec A} W _l,A of the group scheme of Witt vectors of the length l. Moreover we can treat the kernel of the endomorphism of a type F ⁿ + a ₁F^n-1 + … + a _n : W _l,A → W _l,A as our special class, where F is the Frobenius endomorphism and a _k (k = 1, ..., n) are suitable Witt vectors.

This paper deals with sufficient conditions on a poset in order to get it spectral. A motivating question is the following (p. 833 [LO76]): "If X is a height 1 poset such that for all x ≠ y ∈X, ↑ x ∩ ↑y and ↓ x ∩ ↓y are finite, is X spectral?" We obtain the some sufficient conditions for such a poset X to be spectral. In particular, we prove that either if there is a finite subset F ⊆X such that ↓ F ⊇Min X, or if diam X ≤ 2, then the poset X is spectral.

The aim of this paper is to establish necessary and sufficient conditions for certain infinite products generated by Fibonacci numbers and by Lucas numbers to be algebraically independent.

For given group schemes $\mathscr{G}$^{(λ _i )} (i = 1, 2, ...) deforming the additive group scheme G _a to the multiplicative group scheme G _m , T. Sekiguchi and N. Suwa constructed extensions: 0 → $\mathscr{G}$^(λ₂) → $\mathscr{E}$^{(λ₁, λ₂)} → $\mathscr{G}$^(λ₁) → 0, … 0 → $\mathscr{G}$^{(λ _n )} → $\mathscr{E}$^{(λ₁, ..., λ _n )} → $\mathscr{E}$^{(λ₁, ..., λ_n-1)} → 0, … inductively, by calculating the group of extensions Ext¹($\mathscr{E}$^{(λ₁, ..., λ_n-1)}, $\mathscr{G}$^{(λ _n )}). Here changing the group schemes, we treat the group Ext¹($\mathscr{G}$^(λ₀), $\mathscr{E}$^{(λ₁, ..., λ _n )}) of extensions for any positive integers n. The case of n = 2, 3 were studied by D. Horikawa and T. Kondo.