We show that the class of separably closed fields with a generic automorphism is an elementary class, whose theory is model complete in a natural extension of the language of fields with an automorphism. We describe the completions of this theory and obtain some results on types, imaginaries, and modularity.

Austerity is a pointwise algebraic condition on the second fundamental form of an Euclidean submanifold and requires that the nonzero principal curvatures in any normal direction occur in pairs with opposite signs. These submanifolds have been introduced by Harvey and Lawson in the context of special Lagrangian submanifolds.

The main purpose of this paper is to classify all austere submanifolds whose Gauss maps have rank two. This condition means that the image of the Gauss map in the corresponding Grassmannian is a surface. The hypersurface case is due to Dajczer and Gromoll and the three dimensional case to Bryant. We show that any such submanifold is, roughly, a subbundle of the normal bundle of a surface whose ellipse of curvature of a certain order is a circle. We also characterize austere submanifolds which are Kaehler manifolds.

We consider Hausdorff matrices as operators on Hardy spaces of analytic functions. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find conditions on $\mu$ such that the matrix represents a bounded operator. The results unify and extend some known special cases of operators on Hardy spaces such as the Cesàro and generalized Cesàro operators.

We show that the Hausdorff dimension of the Julia set associated to a hyperbolic rational map is bounded away from $2$, where the bound depends only on certain intrinsic geometric exponents. This result is derived via lower estimates for the iterate-counting function and for the dynamical Poincaré series. We deduce some interesting consequences, such as upper bounds for the decay of the area of parallel-neighbourhoods of the Julia set, and lower bounds for the Lyapunov exponents with respect to the measure of maximal entropy.

Kadison's transitivity theorem implies that, for irreducible representations of \cstar algebras, every invariant linear manifold is closed. It is known that CSL algebras have this property if, and only if, the lattice is hyperatomic (every projection is generated by a finite number of atoms). We show several other conditions are equivalent, including the condition that every invariant linear manifold is singly generated. \par We show that two families of norm closed operator algebras have this property. First, let $\LL$ be a CSL and suppose $\AA$ is a norm closed algebra which is weakly dense in $\operatorname{Alg} \LL$ and is a bimodule over the (not necessarily closed) algebra generated by the atoms of $\LL$. If $\LL$ is hyperatomic and the compression of $\AA$ to each atom of $\LL$ is a \cstar algebra, then every linear manifold invariant under $\AA$ is closed. Secondly, if $\AA$ is the image of a strongly maximal triangular AF algebra under a multiplicity free nest representation, where the nest has order type $-\mathbb{N}$, then every linear manifold invariant under $\AA$ is closed and is singly generated.

This paper is devoted to the qualitative study of disconjugate random linear Hamiltonian systems. We relate the principal solutions at $\pm\infty$ with the ergodic structure of the flow, the presence of exponential dichotomy, and the description of the Sacker-Sell spectrum. A continuity theorem for the principal solutions is also provided.

The double decker set $\Gamma$ of a generic map $g:F_0^2\rightarrow M^3$ is the preimage of the singularity of the generic surface $g(F_0)$. If both $F_0$ and $M$ are oriented, then $\Gamma$ is regarded as an oriented 1-cycle in $F_0$, which is shown to be null-homologous if $g(F_0)=0\in H_2(M;{\mathbf Z})$. We also investigate a double decker set of a surface diagram which is a generic surface in ${\mathbf{R}}^3$ with crossing information.

This article uses Cartan-Kähler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kähler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which admit no local isometric Lagrangian immersions into complex three-space. It is expected that isometric Lagrangian immersions of higher-dimensional Riemannian manifolds will almost always be unique. However, there is a plentiful supply of flat Lagrangian submanifolds of any complex $n$-space; we show that locally these depend on $\frac{1}{2}n(n+1)$ functions of a single variable.

We obtain upper bounds for the first eigenvalue of the linearized operator $L_r$ of the $r$-mean curvature of a compact manifold immersed in a space of constant curvature $\delta$. By the same method, we obtain an upper bound for the first eigenvalue of the stability operator associated to $L_r$ when $\delta < 0$.

We show that the $\bar\partial$-Neumann problem is globally regular on a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$ whose closure admits a sufficiently nice Stein neighborhood basis. We also discuss (what turns out to be) a generalization: global regularity holds as soon as the weakly pseudoconvex directions at boundary points are limits, from inside, of weakly pseudoconvex directions of level sets of the boundary distance.

We consider the Schrödinger operator $-\Delta +V$ in $\mathbb{R}^d$ with periodic potential $V$ in the Kato class. We show that, if $d=2$ or $d=3$, the spectrum of $-\Delta +V$ is purely absolutely continuous.

A conjecture of Harbourne and Hirschowitz implies that $r \ge 9$ general points of multiplicity $m$ impose independent conditions to the linear system of curves of degree $d$ when $d(d+3) \ge rm(m+1)-2$. In this paper we prove that the conditions are independent provided $d+2\ge (m+1)(\sqrt{r+1.9}+\pi /8) $.

If $\lambda$ is a vector measure with values in a Banach space and $p > 1$, we consider the space of real functions $L_p(\lambda)$ that are $p$-integrable with respect to $\lambda$. We define two different vector valued dual topologies and we prove several compactness results for the unit ball of $L_p(\lambda)$. We apply these results to obtain new Grothendieck-Pietsch type factorization theorems.

We define an integral of a vector-valued function $f:\Omega\longrightarrow X$ with respect to a bounded countably additive vector-valued measure $\nu:\Sigma\longrightarrow Y$ and investigate its properties. The integral is an element of $X\check{\otimes}Y$, and when $f$ is $\nu$-measurable we show that $f$ is integrable if and only if $\|f\|\in L_{1}(\nu)$. In this case, the indefinite integral of $f$ is of bounded variation if and only if $\|f\|\in L_{1}(|\nu|)$. We also define the integral of a weakly $\nu$-measurable function and show that such a function $f$ satisfies $x^{*}f\in L_{1}(\nu)$ for all $x^{*}\in X^{*}$ and is $|y^{*}\nu|$-Pettis integrable for all $y^{*}\in Y^{*}$.

We investigate differential transcendence properties for a generalized Dirichlet series of the form $\sum_{n=0}^\infty a_n\lambda_n^{-s}$. Our treatment of this series is purely algebraic and does not rely on any analytic properties of generalized Dirichlet series. We establish differential transcendence theorems for a certain class of generalized Dirichlet series. These results imply that the Hurwits zeta-function $\zeta(s,a)$ does not satisfy an algebraic differential equation with complex coefficients.

We present results connecting flatness of extension rings to the Noetherian property for certain intermediate rings between an excellent normal local domain and its completion. We consider conditions for these rings to have Cohen-Macaulay formal fibers. We also present several examples illustrating these results.

A holomorphic function $b$ mapping the unit disk $\disk$ into itself induces a family of measures $\tau_\alpha$, $|\alpha|=1$, on the unit circle $\circle$ by means of Herglotz's Theorem. This family of measures defines the Aleksandrov operator $A_b$ by means of the formula $A_b f(\alpha) = \int_\circle f(\zeta)\,d\tau_\alpha(\zeta)$, at least for continuous $f$. This operator preserves the smoothness classes determined by regular majorants, and is seen to be compact on these classes precisely when none of the measures $\tau_\alpha$ has an atomic part. In the process, a duality theorem for smoothness classes is proved, improving a result of Shields and Williams, and various theorems about composition operators on weighted Bergman spaces are extended to spaces arising from regular weights.

A nonsingular transformation is said to be doubly ergodic if for all sets $A$ and $B$ of positive measure there exists an integer $n>0$ such that $\lambda(T^{-n}(A)\cap A)>0$ and $\lambda(T^{-n}(A)\cap B)>0$. While double ergodicity is equivalent to weak mixing for finite measure-preserving transformations, we show that this is not the case for infinite measure preserving transformations. We show that all measure-preserving tower staircase rank one constructions are doubly ergodic, but that there exist tower staircase transformations with non-ergodic Cartesian square. We also show that double ergodicity implies weak mixing but that there are weakly mixing skyscraper constructions that are not doubly ergodic. Thus, for infinite measure-preserving transformations, double ergodicity lies properly between weak mixing and ergodic Cartesian square. In addition we study some properties of double ergodicity.

This paper explores a Dirichlet type problem on metric measure spaces. The problem is to find a Sobolev-type function that minimizes the energy integral within a class of "Sobolev" functions that agree with the boundary function outside the domain of the problem. This is the analogue of the Euler-Lagrange formulation in the classical Dirichlet problem. It is shown that, under certain geometric constraints on the measure imposed on the metric space, such a solution exists. Under the condition that the space has many rectifiable curves, the solution is unique and satisfies the weak maximum principle.

Some results are established which are useful in determining the order of a finite group with one class of involutions, given knowledge of the centralizer of an involution. An application is provided to illustrate the results. The main tool is a lemma of H. Bender on counting involutions.

It was proved recently that a Banach space fails the Mazur intersection property if and only if the family of all closed, convex and bounded subsets which are intersections of balls is uniformly very porous. This paper deals with the geometrical implications of this result. It is shown that every equivalent norm on the space can be associated in a natural way with a constant of porosity, whose interplay with the geometry of the space is then investigated. Among other things, we prove that this constant is closely related to the set of $\varepsilon$-differentiability points of the space and the set of $r$-denting points of the dual. We also obtain estimates for this constant in several classical spaces.