Let $S$ be a Riemann surface containing at least two punctures $z$ and $z_0$. Let $\mathscr{F}(S)$ be the set of pseudo-Anosov maps of $S$ that are isotopic to the identity on $S\cup \{z\}$. We show that for any $f\in \mathscr{F}(S)$ and any twice punctured disk $\Delta$ enclosing $z$ and $z_0$, the pair $(\partial \Delta, f(\partial \Delta))$ fills $S$, where $\partial \Delta$ denotes the boundary of $\Delta$. Fix such a $\Delta$, and denote by $\mathscr{T}(\Delta)$ the set of twice punctured disks $\Delta'$ on $S$ enclosing $z$ and $z_0$ with the property that $(\partial \Delta, \partial \Delta')$ fills $S$. Let $\Delta_0\in \mathscr{T}(\Delta)$. We describe all possible pseudo-Anosov maps $f$ in $\mathscr{F}(S)$ sending $\Delta$ to $\Delta_0$, and classify elements of $\mathscr{F}(S)$ in terms of $\mathscr{T}(\Delta)$. We also show that there are infinitely many elements $f_k\in \mathscr{F}(S)$ with $f_k(\Delta)=\Delta_0$ such that their dilatations $\lambda(f_k)\rightarrow +\infty$ as $k\rightarrow +\infty$.

In this paper, we give a characterization of the Lefschetz elements in a coinvariant algebra of an arbitrary complex reflection group which contains a binary polyhedral group as a subgroup of index 2. In addition, we examine the relation between the set of the non-strong Lefschetz elements and polyhedra.

In this paper, we study the spectral and pseudospectral properties of the differential operator $H_{\varepsilon} = -\partial_{x}^{2} + x^{2m} +i\varepsilon^{-1}f(x)$ on $L^{2}(\mathbb{R})$, where $\varepsilon > 0$ is a small parameter, $m \in \mathbb{N}$ and $f$ is a real-valued Morse function which satisfies $| \partial^{l}_{x} ( f(x) - |x|^{-k} ) | \le C|x|^{-k-l-1}$ for $l = 0,1,2,3$ and large $|x|$. We show that $\Psi(\varepsilon) = ( \sup_{\lambda \in \mathbb{R} } \| ( H_{\varepsilon} - i\lambda )^{-1} \| )^{-1}$ and $\Sigma(\varepsilon) = \inf \Re (\sigma(H_{\varepsilon}))$ satisfy $C^{-1} \varepsilon^{-\nu(m)} \le \Psi(\varepsilon) \le C \varepsilon^{-\nu(m)}$ and $\Sigma(\varepsilon) \ge C^{-1} \varepsilon^{-\nu(m)}$, $\nu(m) = \min \left\{ \frac{2m}{k+3m+1}, \frac{1}{2} \right\}$. This extends the result of I.~Gallagher, T.~Gallay and F.~Nier [3] (2009) for the case $m=1$ to general $m \in \mathbb{N}$.

We will give a characterization of arithmetical equivalence of number fields in terms of certain associated families of Galois groups with restricted ramification.

For the Riemann zeta-function we consider the fourth moment whose square part admits a shift along the real line. Asymptotic formulas are obtained for the shift parameter around 0 and for the one in unbounded regions.

In this paper we classify a class of surfaces with negative Gaussian curvature parametrized by a generalized Chebyshev net with constant Chebyshev angle in the Euclidean 3-space. As an application we obtain for each constant Chebyshev angle a seven-parameter family of complete surfaces.

We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol $[a_1,a_2, p]$ describes the decomposition law of a prime number $p$ in a certain dihedral extension over $\mathbb{Q}$ of degree 8 determined by $a_1$ and $a_2$. In this paper, we show that the triple symbol $[-p_1,p_2, p_3]$ for certain prime numbers $p_1, p_2$ and $p_3$ can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

We investigate the minimal number of links and knots in embeddings of complete partite graphs in $S^3$. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.

In this paper, we generalize a notion of Koszul resolutions and characterize modules which admits such resolutions. It turns out that for a noetherian ring $A$ and a coherent $A$-module $M$, $M$ has a two dimensional generalized Koszul resolution if and only if $M$ is a pure weight two module in the sense of [HM10]. As an application, we attack the Gersten conjecture for weight two case.

In the present paper we study the interface regularity of the solutions to the differential systems of divergence free and rotation free defined by differential forms in the $N(\geq 3)$-dimensional Euclidean space. Our results are natural extensions of the results of [3] and [5] for $N=3$.

Some inequalities for functions defined by power series concerning two operators in both the noncommutative and commutative case are given. Natural examples for fundamental functions that can be represented by power series are presented as well.

The Fefferman-Stein vector-valued maximal inequalities are established for weighted Orlicz-Morrey spaces. As an application of these inequalities, a framework of weighted Triebel-Lizorkin-Orlicz-Morrey spaces is proposed.

We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.

The class of Riemann zeta distribution is one of the classical classes of probability distributions on $\mathbb{R}$. Multidimensional Shintani zeta function is introduced and its definable probability distributions on $\mathbb{R}^d$ are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on $\mathbb{R}^d$, is also studied.

Let $F(t,x,u,v)$ be a holomorphic function in a neighborhood of the origin of $\mathbb{C}^4$ satisfying $F(0,x,0,0) \equiv 0$ and $(\partial F/\partial v)(0,x,0,0) \equiv 0$; then the equation (A) $t \partial u/\partial t=F(t,x,u, \partial u/\partial x)$ is called a partial differential equation of Briot-Bouquet type with respect to $t$, and the function $\lambda (x)=(\partial F/\partial u)(0,x,0,0)$ is called the characteristic exponent. In [15], it is proved that if $\lambda(0) \not\in (-\infty,0] \cup \{1,2, \ldots \}$ holds the equation (A) is reduced to the simple form $\mbox{(B}_1)$ $t \partial w/\partial t= \lambda (x)w$. The present paper considers the case $\lambda(0)=K \in \{1,2, \ldots \}$ and proves the following result: if $\lambda(0)=K \in \{1,2, \ldots \}$ holds the equation (A) is reduced to the form $\mbox{(B}_2)$ $t \partial w/\partial t= \lambda (x)w+\gamma(x) t^K$ for some holomorphic function $\gamma(x)$. The reduction is done by considering the coupling of two equations (A) and $\mbox{(B}_2)$, and by solving their coupling equations. The result is applied to the problem of finding all the holomorphic and singular solutions of (A).