Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of $(2,5)$-cuspidal edges.

For each $p \gt 0$ we provide the construction of a harmonic function on a homogeneous isotropic tree $T$ in the Bergman space $A^p(\sigma)$ with no finite radial limits anywhere. Here, $\sigma$ is an analogue of the Lebesgue measure on the tree. With the appropriate modifications, the construction yields a function in $A^1(\sigma)$ when $T$ is a rooted radial tree such that the number of forward neighbors increases so slowly that their reciprocals are not summable.

In an $n$-dimensional complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c (\lt 0)$, the horosphere HS, which is defined by ${\rm HS} = \lim_{r\to\infty}G(r)$, is one of nice examples in the class of real hypersurfaces. Here, $G(r)$ is a geodesic sphere of radius $r$ $(0 \lt r \lt \infty)$ in $\mathbb{C}H^n(c)$. The second author ([14]) gave a geometric characterization of HS. In this paper, motivated by this result, we study real hypersurfaces $M^{2n-1}$ isometrically immersed into an $n$-dimensional complex projective space $\mathbb{C}P^n(c)$ of constant holomorphic sectional curvature $c(\gt 0)$.

We study the group of leafwise holomorphic smooth automorphisms of 5-dimensional Reeb components with leafwise complex structure which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we completely determine the structure of the group of leafwise holomorphic automorphisms of such foliations.

Coding theory is connected with number theory via the invariant theory of some specified finite groups and theta functions. Under this correspondence we are interested in constructing, from a combinatorial point of view, an analogous theory of Eisenstein series. For this, we previously gave a formulation of E-polynomials based on the theory of binary codes. In the present paper we follow this direction and supply a new class of E-polynomials. To be precise, we introduce the E-polynomials associated to the $\mathbf{Z}_4$-codes and determine both the ring and the field structures generated by them. In addition, we discuss the zeros of the modular forms obtained from E-polynomials under the theta map.

This paper is the paper announced in [Be2, References [2]]. We show that every compact abelian group of homeomorphisms of $\mathbb{R}^3$ is either zero-dimensional or equivalent to a subgroup of the orthogonal group O(3). We prove a similar result if we replace $\mathbb{R}^3$ by $\mathbb{S}^3$, and we study regular homeomorphisms that are conjugate to their inverses.

We clarify the notion of well-chosen weak solutions of the instationary Navier-Stokes system recently introduced by the authors and P.-Y. Hsu in the article {\em Initial values for the Navier-Stokes equations in spaces with weights in time, Funkcialaj Ekvacioj} (2016). Well-chosen weak solutions have initial values in $L^{2}_{\sigma}(\Omega)$ contained also in a quasi-optimal scaling-invariant space of Besov type such that nevertheless Serrin's Uniqueness Theorem cannot be applied. However, we find universal conditions such that a weak solution given by a concrete approximation method coincides with the strong solution in a weighted function class of Serrin type.

In this work we consider a family of ${\rm Spin}$ complex groups constructed in \cite{anton-article} which have outer automorphisms of order three. We define an action of ${\rm Out}({\rm Spin}(n,\mathbb{C}))\times\mathbb{C}^*$ on the moduli space of ${\rm Spin}$-Higgs bundles and we study the subvariety of fixed points of the induced automorphisms of order three. These fixed points can be expressed in terms of some kind of Higgs pairs associated to certain subgroups of ${\rm Spin}(n,\mathbb{C})$ equipped with a representation of the subgroup. We further the study for the simple case, $G={\rm Spin}(8,\mathbb{C})$.

Let $G$ be a finite nontrivial group and $A(G)$ the Burnside ring of $G$. Let $\mathcal{F}$ be a set of subgroups of $G$ which is closed under taking subgroups and taking conjugations by elements in $G$. Then let $\frak{F}$ denote the category whose objects are elements in $\mathcal{F}$ and whose morphisms are triples $(H, g, K)$ such that $H$, $K \in \mathcal{F}$ and $g \in G$ with $gHg^{-1} \subset K$. Taking the inverse limit of $A(H)$, where $H \in \mathcal{F}$, we obtain the ring $A(\frak{F})$ and the restriction homomorphism ${\rm{res}}^G_{\mathcal{F}} : A(G) \to A(\frak{F})$. We study this restriction homomorphism.