We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$ . In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain ${{\ell }_{1}}$ but none of them is isomorphic to ${{\ell }_{1}}$ . We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$ . In certain cases this ensures that $X$ admits, for each $\alpha \,<\,{{\omega }_{1}}$ , a spreading model ${{\left( \tilde{x}_{i}^{\left( \alpha\right)} \right)}_{i}}$ such that if $\alpha \,<\,\beta $ then ${{\left( \tilde{x}_{i}^{\left( \alpha\right)} \right)}_{i}}$ is dominated by (and not equivalent to) ${{\left( \tilde{x}_{i}^{\left( \beta\right)} \right)}_{i}}$ . Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

We consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the $\text{ADM}$ energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the $\text{ADM}$ energy tends to zero.

In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property ${{N}_{p}}$ associated to ${{K}_{S}}\,\otimes \,{{B}^{\otimes n}}$ where $B$ is base-point free and ample divisor with $B\,\otimes \,{{K}^{*}}\,nef,$ results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

We are interested in Poisson structures transverse to nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $s{{l}_{n}}$ , we construct some families of nilpotent orbits with quadratic transverse structures.

We study closed extensions $\underset{\scriptscriptstyle-}{A}$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted ${{L}_{p}}$ -space. Under suitable conditions we show that the resolvent ${{\left( \lambda -\underset{\scriptscriptstyle-}{A} \right)}^{-1}}$ exists in a sector of the complex plane and decays like $1/\left| \lambda\right|$ as $\left| \lambda\right|\to \infty $ . Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underset{\scriptscriptstyle-}{A}$ .

As an application we treat the Laplace–Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}\,-\,\Delta u\,=\,f,$ $u\left( 0 \right)\,=\,0$ .

Let ${{E}_{/\mathbb{Q}}}$ be an elliptic curve with good ordinary reduction at a prime $p\,>\,2$ . It has a welldefined Iwasawa $\mu $ -invariant $\mu {{\left( E \right)}_{p}}$ which encodes part of the information about the growth of the Selmer group $\text{Se}{{\text{l}}_{{{p}^{\infty }}}}\left( {{E}_{/{{K}_{n}}}} \right)$ as ${{K}_{n}}$ ranges over the subfields of the cyclotomic ${{\mathbb{Z}}_{p}}$ -extension ${{K}_{\infty }}/\mathbb{Q}$ . Ralph Greenberg has conjectured that any such $E$ is isogenous to a curve ${E}'$ with $\mu {{\left( {{E}'} \right)}_{p}}\,=\,0$ . In this paper we prove Greenberg's conjecture for infinitely many curves $E$ with a rational $p$ -torsion point, $p$ = 3 or 5, no two of our examples having isomorphic $p$ -torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum–Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact Kähler manifolds.