Caubel, Nemethi, and Popescu-Pampu in [2] proved that an oriented 3-manifold admits at most one positive contact structure which can be realized as the complex tangency along the link of a complex surface singularity. They call it the Milnor fillable contact structure. Lekili and Ozbagci in [10] showed that a Milnor fillable contact structure is universally tight. In particular, by Honda's classification [5], the link of a cusp singularity is contactomorphic to the positive contact structure associated to the Anosov flow on a Sol-manifold (see [1]). We describe the contact structure on the link of a cusp singularity in two different ways without using Honda's classification theorem. One description is based on the toric method introduced in Mori [15]. The other description is based on Hirzebruch's construction of the Hilbert modular cusps. Consequently, we give certain answers to the problems in Mori [14] concerning the relation between the cusp singularities and the simple elliptic singularities, and the higher dimensional extension of the local Lutz-Mori twist.

When a cyclic group $G$ of prime order acts on a 4-manifold $X$, we prove a formula which relates the Seiberg-Witten invariants of $X$ to those of $X/G$.

We study the inverse Galois problem with restricted ramifications. Let $p$ and $q$ be distinct odd primes. Let $E$ be a non-abelian $p$-group of order $p^3$, and let $k$ be a cyclic extension over ${\mathbf Q}$ of degree $q$. In this paper, we study the existence of unramified extensions over $k$ with the Galois group $E$.

Let $Q$ be a quiver, and $w$ a weight function on the set of arrows of $Q$. In this paper, we will introduce an $R$-algebra ${\sf UD}(Q,w;R)$ over a ring $R$ in which the information how vertices of $Q$ are joined by its arrows with weights should be reflected well. This algebra is obtained by using ${\mathbb Z} Q$-modules where ${\mathbb Z} Q$ is the path algebra of $Q$ over ${\mathbb Z}$. We will particularly focus on quivers and weight functions defined by the subgroup lattice of a finite group $G$, and defined by irreducible characters of subgroups of $G$. The structure of the corresponding ${\mathbb Z}$-algebras ${\sf UD}(Q,w;{\mathbb Z})$ and relations with the group $G$ will be studied.

Let $X$ be a compact connected Riemann surface. Let $\xi_1: E_1\longrightarrow X$ and $\xi_2: E_2\,\longrightarrow X$ be holomorphic vector bundles of rank at least two. Given these together with a $\lambda \in {\mathbb C}$ with positive imaginary part, we construct a holomorphic fiber bundle $S^{\xi_1,\xi_2}_{\lambda}$ over $X$ whose fibers are the Calabi--Eckmann manifolds. We compute the Picard group of the total space of $S^{\xi_1,\xi_2}_{\lambda}$. We also compute the infinitesimal deformations of the total space of $S^{\xi_1,\xi_2}_{\lambda}$. The cohomological Brauer group of $S^{\xi_1,\xi_2}_{\lambda}$ is shown to be zero. In particular, the Brauer group of $S^{\xi_1,\xi_2}_{\lambda}$ vanishes.

We study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of these prolongations, we give explicit expressions of geometric singular solutions of second-order PDEs from the point of view of contact geometry of second order.

While considering some families of polynomials which are orthogonal on a finite or enumerable set of points, Gottlieb was led in the year 1938 to what are now popularly known as the Gottlieb polynomials $\varphi_n(x;\lambda)$. This one-parameter family of polynomials has ever since then been cited widely and investigated extensively in several books, monographs and journal articles. In the present sequel to some of the aforementioned investigations, we introduce and systematically investigate a basic (or $q$-) extension of a multivariable and multiparameter generalization of the Gottlieb polynomials $\varphi_n(x;\lambda)$. We also establish a set of three new families of generating functions for the generalized $q$-Gottlieb polynomials defined here.

The unfolding of a polyhedron along its edges is known as a vertex unfolding if adjacent faces are allowed to be connected not only at an edge but also at a vertex. Demaine et al. [1] showed that every triangulated polyhedron has a vertex unfolding. We extend this result to a tight polyhedron, where a polyhedron is tight if its non-triangular faces are mutually non-incident.

We determine for which $n$, the complete bipartite graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.

In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has $J$-diagonalizable shape operators, then it is homogeneous.

For an arbitrary normed algebra with an approximate identity, we introduce some notions of invertibility of a net in this algebra with respect to the given approximate identity. By means of that, we show a Wiener-Tauberian type theorem in that general situation. An application yields an abstract Wiener-Tauberian theorem for a class of commutative Banach algebras, which implies the classical Wiener-Tauberian theorem for locally compact Abelian groups.

Let $p$, $\lambda$ be real numbers such that $1<p<\infty$, and $0<\lambda<1$. Also let $L^{p,\lambda}(\mathbb{T})$ be Morrey spaces on the unit circle $\mathbb{T}$, and $L^{p,\lambda}_0(\mathbb{T})$ the closure of $C(\mathbb{T})$ in $L^{p,\lambda}(\mathbb{T})$. Zorko [7] gave the predual $Z^{q,\lambda}(\mathbb{T})\ (1/p+1/q=1)$ of $L^{p,\lambda}(\mathbb{T})$. In this article, we show a property of $L^{p,\lambda}_0(\mathbb{T})$ and prove in detail that $L_0^{p,\lambda}(\mathbb{T})$ is the predual of $Z^{q,\lambda}(\mathbb{T})$, whose fact is stated in Adams-Xiao [1].

Let $p$, $\lambda$ be real numbers such that $1\leq p\leq \infty$, and $0\leq\lambda\leq1$. Also we let $L^p(\mathbb{T})$ be the $L^p$-spaces on the unit circle $\mathbb{T}$, $L^{p,\lambda}(\mathbb{T})$ Morrey spaces on $\mathbb{T}$ (cf.~[14]), and $M(L^p,L^{p,\lambda})$ the set of all translation invariant bounded linear operators from $L^p(\mathbb{T})$ to $L^{p,\lambda}(\mathbb{T})$. Figa-Talamanca and Gaudry~[2] showed $M(L^p,L^p)\neq M(L^q,L^q)\ (1<p<q\leq2)$. In this paper, we generalize Gaudry's result. Our main results are $M(L^p,L^{p,\lambda})\neq M(L^q,L^{q,\nu})\ {\rm for}\ \lambda/p\neq\nu/q$ $(1<p, q<\infty,\ 0<\lambda,\nu<1)$, and $M(L^p,L^{p,\lambda})\neq M(L^q,L^{q,\nu})$ for $2<p<q$ and $\lambda/p=\nu/q$\ $(0<\lambda,\nu<1)$. Moreover, we show a relation between $M(L^p,L^{p,\lambda})$ and the measure whose distribution function satisfies a Lipschitz condition (cf.~[4]).

Mahler's method gives algebraic independence results for the values of functions of several variables satisfying certain functional equations under the transformations of the variables represented as a kind of the multiplicative action of matrices with integral entries. In the Mahler's method, the entries of those matrices must be nonnegative; however, in the special case stated in this paper, one can admit those matrices to have a negative entry. We show the algebraic independence of the values of certain functions satisfying functional equations under the transformation represented by such matrices, expressing those values as linear combinations of the values of ordinary Mahler functions.

We give some examples of iterated function systems (IFSs) with overlap on the interval such that the semigroup action they give rise to has a minimal set homeomorphic to the Cantor set.

Let $N$ be a positive integer greater than 1. We define a modular function of level $N$ which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function $j$ generate the modular function field with respect to the principal congruence subgroup of level $N$. Further we study its values at imaginary quadratic points.

For topological spaces $X$ and $Y$, the multiplicity $m(X:Y)$ of $X$ over $Y$ is defined by M. Gromov and K. Taniyama independently. We show that the multiplicity $m(G:\mathbb{R}^1)$ of a finite graph $G$ over the real line $\mathbb{R}^1$ is equal to the cutwidth of $G$. We give a lower bound of $m(G:\mathbb{R}^1)$ and determine $m(G:\mathbb{R}^1)$ for an {\it $n$-constructed graph} $G$.

The purpose of this paper is to give elementary proofs to the theorems due to Avramov on certain determinantal ideals of linear type.