Neumann and Wahl introduced the notion of splice-quotient singularities, which is a broad generalization of quasihomogeneous singularities with rational homology sphere links, and proved the End Curve Theorem that characterizes splice-quotient singularities. The purpose of this paper is to give another proof of the End Curve Theorem. We use combinatorics of “monomial cycles” and some basic ring theory, whereas they applied their theory of numerical semigroups.

We calculate the cuspidal class number of a certain quotient curve of the modular curve $X_0(M)$ with $M$ square-free. For each factor $r$ of $M$, let $w_r$ denote the Atkin-Lehner type involution of $X_0(M)$. Let $M_0$ be a divisor of $M$, and $W_0$ the subgroup of the automorphism group of $X_0(M)$ consisting of all $w_r$ with $r$ dividing $M_0$. Our object is the quotient of $X_0(M)$ by $W_0$. In this paper, we consider the case where $M$ is odd.

We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map $f:X\to Y$, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of $f$. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_p)$ when any tame cyclic extension $N/F$ of degree $p$ has a normal integral basis. We determine all the CM Galois extensions $F/\mathit{Q}$ satisfying $(H_p)$ for the case $p\ge 5$, using the action of the complex conjugation on several objects associated to $F$.

In this paper we deal with hyperbolic operators of second order whose coefficients depend only on the time variable and give necessary conditions and sufficient conditions for the Cauchy problem to be $C^{\infty }$ well-posed. In particular, we give a necessary and sufficient condition (a complete characterization) for $C^{\infty }$ well-posedness when the space dimension is equal to 2 and the coefficients are real analytic functions of the time variable.

We introduce a new class of actions of the group ${\mathit{Z}}^2⋊\mathit{SL}(2,\mathit{Z})$ on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify these actions up to conjugacy and give an explicit description of their centralizers. We also distinguish many of those actions on the AFD ${\mathrm{II}}_1$ factor in view of outer conjugacy.

To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of codimension 2 singularities of link diagrams. For closed braids with a fixed number of strands, trace graphs can be recognized up to equivalence excluding one type of moves in polynomial time with respect to the braid length.

In this paper, we compute Alexander polynomials of a torus curve $C$ of type $(2,5)$, $C:f(x,y)=f_2(x,y{)}^5+f_5(x,y{)}^2=0$, under the assumption that the origin $O$ is the unique inner singularity and $f_2=0$ is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as $C$ is irreducible.

We study the Oseen problem with rotational effect in exterior three-dimensional domains. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces in the whole three-dimensional space. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type and the theory of Calderon-Zygmund kernels in weighted spaces. For the extension of results to the case of exterior domains we use a localization procedure.

Building on the work by J. Jost and C.-J. Xu [32], and E. Barletta et al. [3], we study smooth pseudoharmonic maps from a compact strictly pseudoconvex CR manifold and their generalizations e.g. pseudoharmonic unit tangent vector fields.

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves $E(\lambda ):w^2=z(z-1)(z-\lambda )$. In other words, this formula says that the modular form ${\vartheta }_{00}^4(\tau )$ with respect to the principal congruence subgroup $\mathrm{\Gamma }(2)$ of $\mathit{PSL}(2,\mathit{Z})$ has an expression by the Gauss hypergeometric function $F(1/2,1/2,1;1-\lambda )$ via the inverse of the period map for the family of elliptic curves $E(\lambda )$ (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves $C({\lambda }_1,{\lambda }_2):w^3=z(z-1)(z-{\lambda }_1)(z-{\lambda }_2)$, those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for $C({\lambda }_1,{\lambda }_2)$ is established in [S] and our modular form ${\vartheta }_0^3(u,v)$ (for the definition, see (2.7)) is defined on a two dimensional complex ball , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function $F_1(1/3,1/3,1/3,1;1-{\lambda }_1,1-{\lambda }_2)$.

Let $p\in (0,1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^p({\mathit{R}}^n\times {\mathit{R}}^m)$ to some quasi-Banach space $\mathcal{B}$ if and only if $T$ maps all $(p,2,s_1,s_2)$-atoms into uniformly bounded elements of $\mathcal{B}$. Here $s_1\ge \lfloor n(1/p-1)\rfloor$ and $s_2\ge \lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^p({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $p>1$ or the Hardy space $H^p({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $p\le 1$ but near 1 to the Lebesgue space $L^q({\mathit{R}}^n\times {\mathit{R}}^m)$ with some $q>1$.