Let $T$ be the set of vertices of a tree. We assume that the Green function is finite and $G(s,t) \rightarrow 0$ as $|s| \rightarrow \infty$ for each vertex $t$. For $v$ positive superharmonic on $T$ and $E$ a subset of $T$, the reduced function of $v$ on $E$ is the pointwise infimum of the set of positive superharmonic functions that majorize $v$ on $E$. We give an explicit formula for the reduced function in case $E$ is finite as well as several applications of this formula. We define the minimal fine filter corresponding to each boundary point of the tree and prove a tree version of the Fatou-Naïm-Doob limit theorem, which involves the existence of limits at boundary points following the minimal fine filter of the quotient of a positive superharmonic by a positive harmonic function. We deduce from this a radial limit theorem for such functions. We prove a growth result for positive superharmonic functions from which we deduce that, if the trees has transition probabilities all of which lie between $\delta$ and $1/2-\delta$ for some $\delta \in (0,1/2)$ (for example homogeneous trees with isotropic transition probabilities), then any real-valued function on T which has a limit at a boundary point following the minimal fine filter necessarily has a nontangential limit there. We give an example of a tree for which minimal fine limits do not imply nontangential limits, even for positive superharmonic functions. Motivated by work on potential theory on halfspaces and Brelot spaces, we define the harmonic fine filter corresponding to each boundary point of the tree. In contrast to the classical setting, we are able to show that it is the same as the minimal fine filter.

This paper generalizes a construction of Geramita, Harima, and Shin (Illinois J. Math. \textbf{45} (2001), 1--23). They give an inductive description of a certain set of elements called $n$-type vectors, and use these objects to prove various results about Hilbert functions of sets of points. We extend their notation by inductively describing the monomial ideals in $R$ and identifying certain interesting subsets. We demonstrate that this new notation is useful by using it to calculate multiplicity and the degree of the Hilbert polynomial for quotients of Borel fixed ideals, and by giving another proof of the result of Geramita, Harima, and Shin: The set of $n$-type vectors is in bijective correspondence with all Hilbert functions of finite length cyclic $R$-modules over the polynomial ring $R=\poly{n}$, where $k$ is a field.

For a general nonsingular cubic fourfold $X\subset \PP^5$ and $e\geq 5$ an odd integer, we show that the space $M_e$ parametrizing rational curves of degree $e$ on $X$ is non-uniruled. For $e \geq 6$ an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of $M_e$ is at most one, i.e., passing through a very general point of $M_e$ there is at most one rational curve. For $e < 5$ the spaces $M_e$ are fairly well understood and we review what is known.

We construct a map $\zeta$ from $\K(\mathbb P^d)$ to $(\mathbb Z[x]/x^{d+1})^{\times} \times \mathbb Z$, where $(\mathbb Z[x]/x^{d+1})^{\times}$ is a multiplicative Abelian group with identity $1$, and show that $\zeta$ induces an isomorphism between $\K(\mathbb P^d)$ and its image. This is inspired by a correspondence between Chern and Hilbert polynomials stated in Eisenbud~\cite[Exercise~19.18]{E}. The equivalence of these two polynomials over $\mathbb P^d$ is discussed in this paper.

Let $G$ be a locally compact group and $H$ a discrete amenable subgroup. We prove the existence of a contractive projection $\mathcal{Q}$ of $CV_p(G)$ onto $CV_p(H)$ such that $\operatorname{supp} \mathcal{Q}(T)\subset \operatorname{supp} T$.

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space $\mathcal{H}\, $, and we analyze the set of attainable frame bounds. In the case where $\mathcal{H}\,$ is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite dimensional case is a result we have proven which concerns the decomposition of a positive invertible operator into a strongly converging sum of (not necessarily mutually orthogonal) self-adjoint projections. This decomposition result implies the existence of tight frames in the ellipsoidal surface determined by the positive operator. In the real or complex finite dimensional case, this provides an alternate (but not algorithmic) proof that every such surface contains tight frames with every prescribed length at least as large as $\dim\mathcal{H}\, $. A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.

In 1997, the author proved that the Riemann hypothesis holds if and only if $\lambda_n=\sum [1-(1-1/\rho)^n]>0$ for all positive integers $n$, where the sum is over all complex zeros of the Riemann zeta function. In 1999, E. Bombieri and J. Lagarias generalized this result and obtained a remarkable general theorem about the location of zeros. They also gave an arithmetic interpretation for the numbers $\lambda_n$. In this note, the author extends Bombieri and Lagarias' arithmetic formula to Dirichlet $L$-functions and to $L$-series of elliptic curves over rational numbers.

We consider two natural definitions for the notion of a dynamical system being mixing relative to an invariant sub $\sigma$-algebra $\mathcal{H}$. Both concern the convergence of

\[|E(f\cdot g\circ T^n|\mathcal H)-E(f|\mathcal H)E(g\circ T^n|\mathcal H)|\to 0\]

as $|n|\to \infty$ for appropriate $f$ and $g$. The weaker condition asks for convergence in $L^1$ and the stronger for convergence a.e. We will see that these are different conditions. Our goal is to show that both these notions are robust. As is quite standard we show that one need only consider $g=f$ and $E(f|\mathcal H)=0$, and in this case $|E(f\cdot f\circ T^n|\mathcal H)|\to 0$. We will see rather easily that for $L^1$ convergence it is enough to check an $L^2$-dense family. Our major result will be to show the same is true for pointwise convergence, making this a verifiable condition. As an application we will see that if $T$ is mixing then for any ergodic $S$, $S\times T$ is relatively mixing with respect to the first coordinate sub $\sigma$-algebra in the pointwise sense.

Let $X \subset \R^n$ be a coherent analytic surface. We show that the connected components of global analytic subsets of $X$ are global and we compute the stability index and Bröcker's $t$-invariant of $X$. We also state a real Nullstellensatz for normal surfaces.

In this paper, we study a special class of frame wavelets. The supports of the Fourier transforms of these frame wavelets are the so called frame wavelet sets. Our results provide ways to construct frame wavelets not known previously. In particular, for any given positive number $\alpha$, we show that there exist various frame wavelets whose Fourier transforms are supported by a set of measure $\alpha$.

In this paper we show that if the stability index of $M$ is equal to $n+2$, then the average of the function $|A|^2$ is less than or equal to $n-1$. Moreover, if this average is equal to $n-1$, then $M$ must be isometric to a Clifford minimal hypersurface.

We characterize bijective transformations on the set of all $n$-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.

We investigate the following integral inequality:

\[ \int_D \frac{|u(x)|^p}{\dist(x, D^c)^\alpha} dx \leq c \int_D \!\int_D \frac{|u(x)-u(y)|^p}{|x-y|^{d+\alpha}} dx\,dy, \quad u\in C_c(D), \]

where $\alpha,p>0$ and $D\subset \Rd$ is a Lipschitz domain or its complement or a complement of a point.

The aim of this paper is to motivate the development of a Brunn-Minkowski theory for minimal surfaces. In 1988, H. Rosenberg and E. Toubiana studied a sum operation for finite total curvature complete minimal surfaces in $\mathbb{R}^{3}$ and noticed that minimal hedgehogs of $\mathbb{R}^{3} $ constitute a real vector space [14]. In 1996, the author noticed that the square root of the area of minimal hedgehogs of $\mathbb{R}^{3}$ that are modelled on the closure of a connected open subset of $\mathbb{S}^{2}$ is a convex function of the support function [5]. In this paper, the author

(i) gives new geometric inequalities for minimal surfaces of $\mathbb{R}^{3}$;

(ii) studies the relation between support functions and Enneper-Weierstrass representations;

(iii) introduces and studies a new type of addition for minimal surfaces;

(iv) extends notions and techniques from the classical Brunn-Minkowski theory to minimal surfaces. Two characterizations of the catenoid among minimal hedgehogs are given.

In this paper, as a generalization of Kirillov's orbit theory, we explore the relationship between the dressing orbits and irreducible ${}^*$-representations of the Hopf $C^*$-algebras $(A,\Delta)$ and $(\tilde{A},\tilde{\Delta})$ we constructed earlier. We discuss the one-to-one correspondence between them, including their topological aspects.

On each dressing orbit (which are symplectic leaves of the underlying Poisson structure), one can define a Moyal-type deformed product at the function level. The deformation is more or less modeled by the irreducible representation corresponding to the orbit. We point out that the problem of finding a direct integral decomposition of the regular representation into irreducibles (Plancherel theorem) has an interesting interpretation in terms of these deformed products.

We introduce a condition, called Property -K, which encodes information about the holomorphic structure of fat subdomains. We obtain an equivalence between this condition and the compactness of the $\overline{\partial}$-Neumann operator in any convex domain. We also exhibit a local property of the Kobayashi metric under which the domain is locally a product space.

We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $ \{x\in U:\omega_Y(x)$ contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the persistence of singular-hyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor [GW] and the example in [MPu].

We obtain a number of facts about the proximal and regionally proximal relations in minimal flows, as well as the product of these two relations. The property of the latter relation coinciding with the equicontinuous structure relation is shown to be an invariant of the Ellis group of the flow.

It is a well known fact that every embedded symplectic surface $\Sigma$ in a symplectic four-manifold $(X^4,\omega )$ can be made $J$-holomorphic for some almost-complex structure $J$ compatible with $\omega$. In this paper we investigate when such a structure $J$ can be chosen generically in the sense of Taubes. The main result is stated in Theorem 1.2. As an application of this result we give examples of smooth and non-empty Seiberg-Witten and Gromov-Witten moduli spaces whose associated invariants are zero.

The properties of a bounded derivation $d$ on a complex Banach algebra $A$ such that the spectrum $\sigma([x,dx])$ is finite for all $x\in A$ are studied. In particular we show that, if $\sigma([x,dx])$ is a singleton for each~$x$, then $d$ maps into the radical of $A$.

Every transcendental meromorphic function $f$ in the plane which has only three critical values satisfies

\[ \liminf_{r\to\infty}\frac{T(r,f)}{\log^2r}\geq \frac{\sqrt{3}}{2\pi}, \]

and this estimate is best possible.