Let $R$ be a hyperbolic Riemann surface. Suppose the Teichmüller space $T(R)$ of $R$ is infinite-dimensional. Let $\mu$ be an extremal Beltrami coefficient on $R$ and let $[\mu ]$ be the point in $T(R)$. In this note, it is shown that if $\mu$ is not uniquely extremal, then there exists an extremal Beltrami coefficient $\nu$ in $[\mu ]$ with non-constant modulus. As an application, it follows, as is well known, that there exist infinitely many geodesics between $[\mu ]$ and the base point $[0]$ in $T(R)$ if $\mu$ is non-uniquely extremal.

In this paper we characterize compactness of the canonical solution operator to $\Bar{\partial}$ on weigthed $L^{2}$ spaces on $\mathbb{C}$. For this purpose we consider certain Schrödinger operators with magnetic fields and use a condition which is equivalent to the property that these operators have compact resolvents. We also point out what are the obstructions in the case of several complex variables.

An algorithm is presented for constructing orthogonal multiscaling functions and multiwavelets with higher approximation order in terms of any given orthogonal multiscaling functions. That is, let $\Phi (x) = [\phi _{1}(x), \phi _{2}(x),\ldots , \phi _{r}(x)]^{T} \in (L^{2}(R))^{r}$ be an orthogonal multiscaling function with multiplicity $r$ and approximation order $m$. We can construct a new orthogonal multiscaling function $\Phi ^{new}(x) = [\Phi ^{T} (x), \phi _{r+1}(x), \phi _{r+2}(x),\ldots ,\phi _{r+s}(x)]^{T}$ with approximation order $n(n > m)$. Namely, we raise approximation order of a given multiscaling function by increasing its multiplicity. Corresponding to the new orthogonal multiscaling function $\Phi ^{new}(x)$, orthogonal multiwavelet $\Psi ^{new}(x)$ is constructed. In particular, the spacial case that $r = s$ is discussed. Finally, we give an example illustrating how to use our method to construct an orthogonal multiscaling function with higher approximation order and its corresponding multiwavelet.

In this paper we consider the quadratic form $\phi [x] = \sum _{i=1}^{n} x_{i}^{2}$ over the vector space $\mathbb{Q}_{n}^{1}$. We take a $\mathbb{Z}$-maximal lattice $L$ in $\mathbb{Q}_{n}^{1}$ with respect to $\phi$. Let $\{L^{(i)}\}_{i=1}^{k(n)}$ be a complete set of representatives for the classes belonging to the genus of $L$. Applying Shimura's mass formula, we determine these representatives $L^{(i)}$ explicitly for $n = 11, 13$, and $14$. Consequently we obtain class numbers $k(11) = 3$, $k(13) = 4$, and $k(14) = 4$.

The basin problem for a strange attractor asks the asymptotic distribution of Lebesgue almost every initial point in the basin of attraction. A solution to this problem for Hénon-like attractors was initially given by Benedicks-Viana, and later by Wang-Young, under certain assumptions on the Jacobian of the map, which are used in a crucial way to control the growth of volumes under iteration. The purpose of this paper is to remove the assumption on the Jacobian in their solutions, in a hope that the argument can be extended to a broader class of Hénon-like maps which are not necessarily invertible and possess singularities.

We give a classification of $C^{2}$-regular and non-degenerate projectively Anosov flows on three-dimensional manifolds. More precisely, we prove that such a flow must be either an Anosov flow or represented as a finite union of $\mathbb{T}^{2} \times I$-models.

Let $R$ be a (not necessary finite dimensional) commutative noetherian ring and let $C$ be a semi-dualizing module over $R$. There is a generalized Gorenstein dimension with respect to $C$, namely $\mathrm{G}_{C}$-dimension, sharing the nice properties of Auslander’s Gorenstein dimension. In this paper, we establish the Faltings’ Annihilator Theorem and it’s uniform version (in the sense of Raghavan) for local cohomology modules over the class of finitely generated $R$-modules of finite $\mathrm{G}_{C}$-dimension, provided $R$ is Cohen-Macaulay. Our version contains variations of results already known on the Annihilator Theorem.

In this paper we introduce some new sequence spaces combining a lacunary sequence, an infinite matrix, a bounded sequence and an Orlicz function. We discuss some topological properties and establish some inclusion relations between these spaces. It is also shown that if a sequence is lacunary $A_{p}$-convergent with respect to an Orlicz function then it is lacunary strongly $S^{\theta}(A)$-statistically convergent.

Here we prove the generalization of Weinstein’s theorem for Finsler manifolds: an isometry of a compact oriented Finsler manifold of positive flag curvature has a fixed point supposed that it preserves the orientation of the manifold if its dimension is even, or reverses it if odd.

We generalize Borcherds’ construction of infinite products to higher level vector valued modular forms of Nebentypus. Then we obtain meromorphic modular functions for the orthogonal group whose zeros or poles lie on the Heegner divisors. The construction involves twisted Siegel theta functions and the singular theta integral.

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity $\lambda |u|^{2}u$, where $\lambda$ is a complex constant satisfying $\mathrm{Im}\lambda < 0$, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as $t \to +\infty$ in a suitable function space. Our modified free dynamics decays like $(t\log t)^{-1/2}$ as $t\to \infty$.