In this note we establish the validity, in the case of unipotent group schemes over non-perfect fields, of an analog of Serre’s conjectures for algebraic groups, which relates properties of Galois (or flat) cohomology of unipotent group schemes to finite extensions of non-perfect fields. We also establish an interesting property of Russell’s defining equations of connected smooth one-dimensional unipotent groups over a field $k$.

We give the first complete classification of free and non-free multiplicities on an arrangement, called the deleted $A_{3}$ arrangement, which admits both of them.

We study many-term relations for multiple Dedekind sums including the case generalized by means of Dirichlet characters. We mainly use the method due to Carlitz. The main results contain the original reciprocity formula for Dedekind sums and its generalized formulas due to Apostol, Snyder and Carlitz.

A holomorphic action of a Lie group $G$ on a complex manifold $D$ is called \textit{strongly visible} if there exist a totally real submanifold $S$ which meets every $G$-orbit in $D$ and an anti-holomorphic diffeomorphism $\sigma $ such that $\sigma |_{S}=\mathop{\mathrm{id}}_{S}$ and $\sigma$ preserves every $G$-orbit. In this paper, we prove that Kac’s multiplicity-free space is strongly visible, that is, if $(G_{\mathbf{C}},V)$ is an irreducible multiplicity-free space of a complex reductive Lie group $G_{\mathbf{C}}$, then a compact real form of $G_{\mathbf{C}}$ acts on $V$ in a strongly visible fashion. Furthermore, we give an explicit description of the choice of a totally real submanifold $S$ and an anti-holomorphic involution $\sigma$. This gives an evidence to Kobayashi’s conjecture , that is, $\dim_{\mathbf{R}}S$ coincides with the rank of the polynomial representation of $G_{\mathbf{C}}$ on $\mathbf{C}[V]$ in this setting.

A parameterized family of continuous functions which was considered by the first author is re-visited in the case when they are monotonically increasing. We prove that the functions are not only continuous and strictly increasing but also singular, i.e., their derivatives are zero almost everywhere.

We study the unique continuation property of some second order elliptic systems.

In this paper, we deal with the problem of uniqueness of meromorphic functions that share three values IM and a fourth value CM, and prove some results which answer a open question on uniqueness of meromorphic functions. Examples show that the conditions of theorems in this paper are necessary.

In this paper, we consider the Royden compactifications relative to $p$-Dirichlet integrals of infinite graphs and noncompact Riemannian manifolds, and study the behavior of rough isometries in the compactifications, proving bijective correspondence of the spaces of $p$-harmonic functions with finite $p$-energy.

The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $\mathfrak{g}$ where $\mathfrak{g}$ is $\mathfrak{gl}_{\infty}$ or the affine Lie algebra $A^{(1)}_{\ell}$, and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type B. In this note, we present similar conjectures for certain classes of irreducible representations of affine Hecke algebras of type D. The crystal for type D is a double cover of the one for type B.