We investigate the pairs $(X,\,\varepsilon )$ consisting of a smooth complex projective variety $X$ of dimension $n$ and an ample vector bundle $\varepsilon $ of rank $n\,-\,1$ on $X$ such that $\varepsilon $ has a section whose zero locus is a smooth elliptic curve.

Let $B\,=\,{{B}_{n}}$ be the open unit ball of ${{C}^{n}}$ with volume measure $v$, $U\,=\,{{B}_{1}}$ and $B$ be the Bloch space on $U.\,{{\mathcal{A}}^{2,\alpha }}(B)$, $1\,\le \,\alpha \,<\infty $, is defined as the set of holomorphic $f\,:\,B\,\to \,C$ for which

$$\int_{B\,}{{{\left| f(z) \right|}^{2}}}{{\left( \frac{1}{\left| z \right|}\,\log \frac{1}{1\,-\left| z \right|} \right)}^{-\alpha }}\,\frac{dv(z)}{1\,-\,\left| z \right|}\,<\,\infty $$

if $0\,<\,\alpha \,<\infty $ and ${{\mathcal{A}}^{2,1}}(B)\,=\,{{H}^{2}}(B)$, the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic $f\,:\,B\,\to \,U$ for which the composition operator ${{C}_{f}}\,:\,B\,\to \,{{\mathcal{A}}^{2,\alpha }}(B)$ defined by ${{C}_{f}}(g)=g\,\text{o}\,f\text{,}\,g\in B$, is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

A preordering $T$ is constructed in the polynomial ring $A=\mathbb{R}[{{t}_{1}},{{t}_{2}},...]$ (countablymany variables) with the following two properties: (1) For each $f\,\in \,A$ there exists an integer $N$ such that $-\,N\,\le \,f\left( p \right)\,\le \,N$ holds for all $P\in \text{Spe}{{\text{r}}_{T}}(A)$. (2) For all $f\,\in \,A$, if $N+f,N-f\in T$ for some integer $N$, then $f\,\in \,\mathbb{R}$. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering $T$ in a finitely generated $\mathbb{R}$-algebra $A$, if property (1) holds, then for any $f\in A,f>0\,\text{on}\,\text{Spe}{{\text{r}}_{T}}(A)\Rightarrow f\in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\sum{{{C}^{2}}}$ (the set of sums of squares) as the preordering.

Bessa [Be] proved that for given $n$ and ${{i}_{0}}$, there exists an $\varepsilon (\text{n,}\,{{i}_{0}})\,>\,0$ depending on $n$, ${{i}_{0}}$ such that if $M$ admits a metric $g$ satisfying $\text{Ri}{{\text{c}}_{(M,g)}}\ge n-1,\text{in}{{\text{j}}_{(M,g)}}\ge {{i}_{0}}\,>\,0$ and $\text{dia}{{\text{m}}_{(M,g)}}\,\ge \,\pi \,-\,\varepsilon $, then $M$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.

We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors $({{x}_{\text{n}}})$ to be eigenvectors $T{{x}_{n}}={{\lambda }_{n}}{{x}_{n}}({{\lambda }_{n}}\ne {{\lambda }_{k}}\text{for}\,n\ne k)$ of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1) for the sequence ${{\left( {{x}_{n}}+{{\epsilon }_{n}}{{v}_{n}} \right)}_{n\ge k(\epsilon )}}$ to be admissible for every admissible $({{x}_{\text{n}}})$ and for a suitable choice of small numbers ${{\epsilon }_{n}}\,\ne \,0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist ${{\text{ }\!\!\gamma\!\!\text{ }}_{n}}\,\in \,C$ such that ${{v}_{n}}\,=\,{{\gamma }_{n}}{{v}_{k}}\,\text{for}\,n\,\ge \,\text{k}$ (Theorem 2); (2) for a bounded operator $A$ to transform admissible families $({{x}_{\text{n}}})$ into admissible families $(A{{x}_{n}})$ it is necessary and sufficient that $A$ be left invertible (Theorem 4).

In this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and $\left( t,\,m,\,s \right)$-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (i.e., necessary conditions) for existence of these objects.

We study boundedness properties of the $q$-mean-square operator {{S}^{(q)}} on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2\,\le \,q\,<\,\infty $. We establish that a.s. finiteness of ${{S}^{(q)}}$ for every bounded $E$-valued analytic martingale implies strong $(p,\,p)$-type estimates for ${{S}^{(q)}}$ and all $p\,\in \,(0,\,\infty )$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function ${{S}^{(q)}}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly $\text{PL}$-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the ${{L}^{p}}$-boundedness of the usual square-function on scalar-valued analytic martingales.

For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^{*}$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

Let $Q$ be a simple Artinian ring with finite dimension over its center. An order $R$ in $Q$ is said to be Prüfer if any one-sided $R$-ideal is a progenerator. We study prime and primary ideals of a Prüfer order under the condition that the center is Prüfer. Also we characterize branched and unbranched prime ideals of a Prüfer order.

Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, e.g., [1], [2], [4], [7], [11] and [12]. It is therefore all the more surprising that the unpleasant properties of these invariants may be derived from two unexpected, and remarkable, nice properties: namely for vector permutation invariants of the cyclic group $Z/p$ of prime order in characteristic $p$ the image of the transfer homomorphism $\text{T}{{\text{r}}^{Z/p}}\,:\,F[V]\,\to \,F{{[V]}^{Z/p}}$ is a prime ideal, and the quotient algebra $F{{[V]}^{Z/p}}/\,\text{IM(T}{{\text{r}}^{Z/p}})$ is a polynomial algebra on the top Chern classes of the action.

Let $R$ be a one-dimensional locally analytically irreducible Noetherian domain with finite residue fields. In this note it is shown that if $I$ is a finitely generated ideal of the ring $\text{Int(}R)$ of integer-valued polynomials such that for each $\text{x}\,\in \,R$ the ideal $I\text{(}x\text{)}=\{f(x)|f\in I\}$ is strongly $\text{n}$-generated, $n\,\ge \,2$, then $I$ is $\text{n}$-generated, and some variations of this result.

A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

Let $(X,\,\left\| \,.\, \right\|)$ be a Minkowski space (finite dimensional Banach space) with unit ball $B$. Various definitions of surface area are possible in $X$. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

A separator of a connected graph $G$ is a set of vertices whose removal disconnects $G$. In this paper we give various conditions for a separator to contain a minimal one. In particular we prove that every separator of a connected graph that has no thick end, or which is of bounded degree, contains a minimal separator.

Let $H$ be an exceptional, adjoint group of type ${{E}_{6}}$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $P{{D}^{\times }}\,\times \,{{G}_{2}}\left( k \right)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of ${{G}_{2}}\left( k \right)$.

In this paper we show that the Lie-group $\text{Pi}{{\text{n}}_{4}}$ is isomorphic to the semidirect product $\text{(S}{{\text{U}}_{2}}\times \text{S}{{\text{U}}_{2}})\text{Z/2}$ where $Z/2$ operates by flipping the factors. Using this structure theorem we prove a classification theorem for $\text{Pi}{{\text{n}}_{4}}$-bundles over a finite 4-complex $X$.

Let $R$ be a non-$\text{GPI}$ prime ring with involution and characteristic $\ne 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta \,=\,\rho \,+\,\varepsilon$ where $\varepsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $\left[ K,\,K \right]$, and $\rho$ can be extended to an ordinary derivation of $\left\langle K \right\rangle$ into $RC$, the central closure.

A theorem of Korányi and Wolf displays any Hermitian symmetric domain as a Siegel domain of the third kind over any of its boundary components. In this paper we give a simple proof that an analogous realization holds for any self-adjoint homogeneous cone.