Let $f: X \to Y$ be a surjective holomorphic map of compact complex manifolds and $\Delta$ the degeneracy locus of $f$. In this paper we shall discuss relationship between infinitesimal deformations of $f$ and the corresponding infinitesimal displacements of $\Delta$ in $Y$. We shall prove that two kinds of Kodaira-Spencer maps are compatible under certain assumptions. As an application of our main theorem, deformations of quadric bundles shall be discussed.

A bounded operator $T\in L(X)$, $X$ a Banach space, is said to satisfy Weyl's theorem if the set of all spectral points that do not belong to the Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues and having finite multiplicity. In this article we give sufficient conditions for which Weyl's theorem for an extension $\overline T$ of $T$ (respectively, for $T$) entails that Weyl's theorem holds for $T$ (respectively, for $\overline T$).

We generalize Osburn's work ([6]) about a congruence for traces defined in terms of Hauptmoduli associated to certain genus zero groups of higher levels.

We define and study the Bessel potential and inhomogeneous Besov spaces associated with the Dunkl operators on $\mathbf{R}^d$. As applications on these spaces we construct the Sobolev type embedding theorem and the paraproduct operators associated with the Dunkl operators, as similar to those defined by Bony. We also establish Strichartz type estimates for the Dunkl-Schrödinger equation and finally we study the problem of well posedness of the generalized heat equation.

We prove the existence of\ positive solutions for the $p$-Laplacian system \[ \left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in}~\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in}~\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \text{on}~\partial \Omega \,, \end{array} \right. \] where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda $ is a large parameter.

In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation formulas for double $L$-values.

About fifty years ago, questions on the existence and non-existence of finite invariant measures were studied by various authors and from different directions. In this article, we examine these classical results and prove directly that all the conditions introduced by these authors are equivalent to each other. We begin at the fundamental level of a recurrent transformation whose properties can be strengthened to obtain the aforementioned classical results for the existence of a finite invariant measure. We conclude with the introduction of a new property, Strongly Weakly Wandering (sww) sequences, the existence of which is equivalent to the non-existence of a finite invariant measure. It is shown that every sww sequence is also an Exhaustive Weakly Wandering (eww) sequence for ergodic transformations. Although all ergodic transformations with no finite invariant measure are known to have eww sequences, there are exceedingly few actual examples for which explicit eww sequences can be exhibited. The significance of sww sequences is that it provides a condition which is easier to verify than the condition for eww sequences (Proposition 4.5). In a second paper, we will continue these studies and also connect them to some of the more recent derived conditions for finite invariant measures. The impetus for this work, began with the late Professor Shizuo Kakutani, with whom the authors worked and had many fruitful discussions on these topics.

In this paper we consider surfaces with negative Gaussian curvature parametrized by a generalized Chebyshev net with constant Chebyshev angle in the Euclidean 3-space. We characterize these surfaces in terms of a meromorphic function which satisfies a certain differential equation. Moreover, we show that these surfaces have the geometric property that the asymptotic lines have the same sign of geodesic curvature. As an application we obtain for each constant Chebyshev angle a four-parameter family of complete surfaces.

In [9], Boyer and Galicki introduced a contact reduction method in the context of Sasakian manifolds, which produces 5-dimentional Sasaki-Einstein manifolds from a 7-sphere. In this paper, we compute very explicitly the metric obtained from the above mentioned reduction via a projection, $S^3 \times S^3 \to S^2 \times S^3$, and show that this metric is the homogeneous Kobayashi-Tanno metric.

Moen introduced a multiple weights class. Moen proved that the multiple weights condition of vector of weights implies each weight function satisfies a certain $A_{p}$ condition. In 2010, Chen and Xue improved sufficency under an unnatural condition. However we can remove this condition and prove necessity. We also give a typical example of the multiple weights class.

Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called $d$-preserving maps between subsets of groups and apply them in several directions. We consider $d$-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group $\mathbb T$. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.

Suppose that $L/k$ is a finite and abelian extension such that $k$ is a totally real base field and $L$ is a CM-field. We regard the ideal class group $\mathrm{Cl}_{L}$ of $L$ as a $\mathrm{Gal}(L/k)$-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for $L/k$ times the annihilator ideal of the roots of unity in $L$ is in the Fitting ideal of $\mathrm{Cl}_{L}$, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual $(\mathrm{Cl}_{L})^{\vee}$. We systematically construct extensions $L/k$ for which these properties do not hold, and also give numerical examples.

Let $a$ be a positive integer with $a\geq 2$ and $Q_a(k,l)$ be the set of odd prime numbers $p$ such that the residual order of $a$ in $\mathbf{Z}/p\mathbf{Z}^\times$ is congruent to $l \bmod k$. The natural density of the set $Q_a(q,0)$ ($q$ is a prime) is already known. In this paper, we consider the set $S_{a,b}(k,l)$, which consists of the primes $p$ that belong to $Q_a(k,l)$ and satisfy $\big(\frac{b}{p}\big)=1$, where $\big(\frac{b}{p}\big)$ is the Legendre symbol and $b$ is a fixed integer. Heuristically, the natural density of $S_{a,b}(k,l)$ is expected to be half of that of $Q_a(k,l)$, but it is not true for some choices of $a$ and $b$. In this paper, we determine the natural density of $S_{a,b}(k,l)$ for $(k,l)=(2,j), (q,0), (4,l)$, where $j=0,1$, $q$ is an odd prime and $l=0,2$.

It is known that there are exactly 95 weighted projective spaces having Gorenstein $K3$ surfaces as anticanonical divisors, some of which have isometric Picard lattices for generic members. For each set of such families, an explicit birational correspondence coming from a torus action is constructed in this paper. As a result the number of `essentially different' families of weighted Gorenstein $K3$ surfaces is 75.

Let $S$ be a Riemann surface of finite area with at least one puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, the pair $(a, f^m(a))$ of geodesics fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint, then $(a,f^m(a))$ fills $S$ for any $m\geq 2$.

We show that the minimal number of singular fibers in a genus-2 Lefschetz fibration over a closed surface of genus $h$ is equal to 5 if $h\geq 3$, 5 or 6 if $h=2$ and 6 or 7 if $h=1$.

A classification of complete Kaehler submanifolds $M_{n}$ in $P_{n+p}(C)$ with scalar curvature $\rho > n^{2}$ is given, resolving a conjecture of K. Ogiue.

In the present note, we will show that there are infinitely many composite twisted torus knots.

In this paper we classify all non-trivial Riemannian submersions with connected fibers from any of the simply connected, rank-one projective spaces. The result follows from results of Gromoll, Grove, Wilking, Becker, Casson, Gottlieb, Schultz, Ucci, and Wolf, together with results of the author.

In this paper, we study the kernel of the reciprocity map of certain simple normal crossing varieties over a finite field and give an example of a simple normal crossing surface whose reciprocity map is not injective for any finite scalar extension.