Jeffrey and Kirwan gave expressions for intersection pairings on the reduced space ${{M}_{0\,}}=\,{{\mu }^{-1}}(0)/G$ of a Hamiltonian $G$ -space $M$ in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian $\text{SU(2)}$ -space. The definition of quasi-Hamiltonian $G$ -spaces was introduced by Alekseev, Malkin and Meinrenken.

Let $N$ be an integer which is larger than one. In this paper we study invariant subspaces of ${{L}^{2}}({{\mathbb{T}}^{N}})$ under the double commuting condition. A main result is an $N$ -dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an $N$ -dimensional version of Lax's theorem.

We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$ . In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\mathbb{Z}$ there will be $\mathbb{Q}$ -linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$ .

In this paper we establish a Hopf type lemma and a $\text{CR}$ type inversion for the generalized Greiner operator. Some nonlinear Liouville type results are given.

This article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

Let $\mathcal{A}$ be a ${{C}^{*}}$ -algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\,\mathcal{A}\,\to \,E$ , the composition $j\,\circ \,T$ factors through a diagonal operator from ${{l}^{2}}$ into ${{l}^{1}}$ . In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2\,<\,p\,<\,\infty $ , any absolutely summing operator from $\mathcal{A}$ into $E$ factors through a diagonal operator from ${{l}^{p}}$ into ${{l}^{2}}$ .

It is known that any non-archimedean Fréchet space of countable type is isomorphic to a subspace of $c_{0}^{\mathbb{N}}$ . In this paper we prove that there exists a non-archimedean Fréchet space $U$ with a basis $({{u}_{n}})$ such that any basis $({{x}_{n}})$ in a non-archimedean Fréchet space $X$ is equivalent to a subbasis $({{u}_{kn}})$ of $({{u}_{n}})$ . Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of $U$ . In contrast to this, we show that a non-archimedean Fréchet space $X$ with a basis $({{x}_{n}})$ is isomorphic to a complemented subspace of $c_{0}^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: ${{c}_{0}},\,{{c}_{0}}\,\times \,{{\mathbb{K}}^{\mathbb{N}}},\,{{\mathbb{K}}^{\mathbb{N}}},\,c_{0}^{\mathbb{N}}$ . Finally, we prove that there is no nuclear non-archimedean Fréchet space $H$ with a basis $({{h}_{n}})$ such that any basis $({{y}_{n}})$ in a nuclear non-archimedean Fréchet space $Y$ is equivalent to a subbasis $({{h}_{kn}})$ of $({{h}_{n}})$ .

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\mathbb{R}$ or $\mathbb{C}$ .

In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m\,>\,4$ , and we show that any set of Yang-Mills connections with the uniformly bounded ${{L}^{\frac{m}{2}}}$ -norm of curvature is compact in ${{C}^{\infty }}$ topology.

Exponent information is proven about the Lie groups $SU(3),\,SU(4),\,Sp(2)$ , and ${{G}_{2}}$ by showing some power of the $H$ -space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8,\,32,\,64$ , and ${{2}^{8}}$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of 2 for $SU(4)$ and $Sp(2)$ , and off by at most two powers of 2 for ${{G}_{2}}$ .

The stable basin theorem was introduced by Basmajian and Miner as a key step in their necessary condition for the discreteness of a non-elementary group of complex hyperbolic isometries. In this paper we improve several of Basmajian and Miner’s key estimates and so give a substantial improvement on the main inequality in the stable basin theorem.

We prove that a class of perturbations of standard $\text{CR}$ structure on the boundary of threedimensional complex ellipsoid ${{E}_{p,\,q}}$ can be realized as hypersurfaces on ${{\mathbb{C}}^{2}}$ , which generalizes the result of Burns and Epstein on the embeddability of some perturbations of standard $\text{CR}$ structure on ${{S}^{3}}$ .

We prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in ${{L}^{1}}$ -norm.

For a locally compact group $G$ , the convolution product on the space $N({{L}^{p}}\ (G))$ of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra $N({{L}^{p}}\ (G))$ and relate them to some properties of the group $G$ , such as compactness, finiteness, discreteness, and amenability.

In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for ${{C}^{*}}$ -algebras on Hardy submodules.

Voiculescu has previously established the uniqueness of the wave operator for the problem of ${{\mathcal{C}}^{(0)}}$ -perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal $\mathcal{C}$ has the property ${{\lim }_{n\,\to \,\infty }}\,{{n}^{-1/2}}\,\left\| {{P}_{n}} \right\|\mathcal{C}\,=\,0$ , where $\{{{P}_{n}}\}$ is any sequence of orthogonal projections with rank $({{P}_{n}})\,=\,n$ . We prove that the same uniqueness result holds true so long as $\mathcal{C}$ is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

In this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.

We investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.