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The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors

This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 > p > ∞ ) (1>p>\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

Let Ω C ℝ N be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)w): x ∈ ℝ, ω ∈ Ω} with r > 0 and smooth, and the natural metric ds 2 = (1 + r'(x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends: r(x) = |x| + O(x -1 ) as x → ±∞. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrodinger evolution e itΔ M and the wave evolution e it √ -Δ M are obtained for data of the form f (x, ω) = y n (ω)u(x), where Y n are eigenfunctions of -Δ Ω with eigenvalues u 2 n . In this paper we discuss all cases d + n > 1. If n ≠ 0 there is the following accelerated local decay estimate: with 0 1, ∥ω σ e itΔM Y n f ∥ L ∞ (M) ≤ C (n, M, σ) t - d+1 / 2 -σ ∥w -1 σ f ∥ L 1 (M) , where w σ (x) = 〈x〉 -σ , and similarly for the wave evolution. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrodinger operators of the form -∂ 2 ξ + (v 2 - 1 / 4 )〈ξ〉 -2 + U(ξ) on the line where U is real-valued and smooth with U (l) (ξ) = 0( ξ-3-l ) for all l ≥ 0 as ξ → ±∞ and v > 0. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case. (B) Estimation of oscillatory integrals by (non)stationary phase.

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use "group chunks" or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.

The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of $L^2$ jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for any weight order greater than $-1$ in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions., The title was changed from "Weighted Bergman spaces of domains with Levi-flat boundary: geodesic segments on compact Riemann surfaces". 25 pages, final version, to appear in Transactions of the American Mathematical Society

In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series E 6 ( τ ) E_6(\tau ) . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

JOSE MANUEL MARCO AND JAVIER PARCET´Abstract. In this paper we present a unified approach to the spectral analysisof an hypergeometric type operator whose eigenfunctions include the classicalorthogonal polynomials. We write the eigenfunctions of this operator by meansof a new Taylor formula for operators of Askey-Wilson type. This gives rise tosome expressions for the eigenfunctions, which are unknown in such a generalsetting. Our methods also give a general Rodrigues formula from which severalwell known formulas of Rodrigues type can be obtained directly. Moreover,other new Rodrigues type formulas come out when seeking for regular solutionsof the associated functional equations. The main difference here is that, incontrast with the formulas appearing in the literature, we get non-ramifiedsolutions which are useful for applications in combinatorics. Another fact,that becomes clear in this paper, is the role played by the theory of ellipticfunctions in the connection between ramified and non-ramified solutions.

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

Let LR(H) denote the set of all closed linear relations on a Hilbert space H (which contains all closed linear operators on H). In this paper, for every E E LR(H) we define and study two associated linear operators on H, cos(E) and sin(E), which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of E, a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the cos(E) and sin(E) operators to operator theory will be given in a forthcoming paper.

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

We study geometric properties of the metric projection r: M S of an open manifold M with nonnegative sectional curvature onto a soul S. ir is shown to be C? up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M, and stu(dy the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M. The resolution of the Soul conjecture of Cheeger and Gromoll by Perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. Orne of the key results in [14] is that the metric projection wF: M -> S which maps a point p in M to the point 7r(p) in S that is closest to p is a Riemannian submersioni. Perelman also observed that this map is at least of class C1, and later it was shown in [9] that 7w is at least C2, and C? at almnost every point. The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M inust be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map -w. The paper is essentially structured as follows: After introducing notation and recalling some basic results in sectioni 1, we establish in section 2 the existence of maps similar to -w for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle vti(S) admits a metric with nonnegative sectional curvature. As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound (see also [81). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach maanifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us. Properties such as these are established via stanidard submersion techniques, but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of M far away from the soul. One way around Received by the editors August 18, 1997. 1991 Mathematics Subject Classification. Primary 53C20.

In this paper, we study the topology of spaces of n n -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the E 2 E^2 term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163–221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

We study the elliptic equation Δ u + k u − K u p = 0 \Delta u + ku - Ku^{p} = 0 on complete noncompact Riemannian manifolds with K K nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space R n {\mathbf {R}}^{n} and the hyperbolic space H n {\mathbf {H}}^{n} are carried out when k k is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.

If ¢ is a generic cubic rnetaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to X is an Euler product of degree 4 that has a functional equation and rnerornorphic continu.ation to the whole cornplex plane. This correspondence is obtained by convolving (b with the cubic 0-function on GL(3) in a Shirnura type RankinSelberg integral. 0. INTRODUCT[ON Suppose 0 is a metaplectic automorphic form of minimal level on the 3-fold cover of GSp(4) that is an eigenfunction of all the Hecke operators. If 0 has any non-zero Whittaker coefficients, then 0 is called generic. In this case, this paper will show tha1; there is a Dirichlet series in the Whittaker coefficients of 0 that has a formula1;ion as a degree 4 Euler product. Moreover, this Euler product has a meromorphic continuastion to the whole complex plane. This association of the Euler product with 0 will be obtained via a Shimura type JRankin-Selberg integral involving 0 and a H-function on the 3-fold cover of GL(3). Historically) the problem of associating an Euler product which has meromorphic continuation and functional equation with a metaplectic automorphic form originated with the work of Shimura [Shi]. More specifically, suppose f (z). = E a(n) qn is a holomorphic modular form of half-integral weight k/2, which is an eigenform of the Hecke operators Tp2, i.e. Tp2 f = )pf. Then via a JRankin-Selberg integral of the form f __ (0.1) | f (z) H(z) E(z, s) dz, where H(z) is a classical theta function and E(z, s) is an integral weight Eisenstein series, Shimura obtains an Euler product of the form (0.2) t| (1 _ Ap ps + pk-2-2s)-1 p The analytic continuation and functional equation of this Euler product follow from the siirlilar properties of E(z, s) in (0. 1) . Bump and Hoffstein [BH2] have subsequently extended these techniques of [Shig to GL(3) by finding a Rankin-Selberg integral of a metaplect1c automorphic form on the 3-fold cover of GL(3) which produces an Euler product of degree 3. Just as in [ShiX, this Euler product is shown to have meromorphic continuation to the whole Received by the editors November 14, 1995 and, in revised forrn, May 21, 1996. 1991 Mathematics S?lbject Classification. Prirnary llF55, llF30. @1998 Ameri( an Mathematical Society 975 This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms THOMAS GOETZE 976 complex plane and to have a functional equation under s 1-s. The integral which represents this Euler product involves the 0-function on the 3-fold cover of GL(3) over the field Q (e21rt/3) ? wllich has been studied independently by Proskurin [Pr] and Bump and HofEstein [BH12. In addition, Bump and HofEstein [BH2] have conjectured that Euler products with meromorphic continuation and functional equation may be obtained by convolving metaplectic automorphic forms on the n-fold cover of GL(r) against H-functions on the n-fold cover of GL(n). This was carried out in [BH3] in the case r-2 and n > 2. Friedberg and Wong [FrW] have also used Shimura's method to associate an Euler product to a generic metaplectic automorphic form on the double cover of the symplectic group GSp(4). They have found an integral (inspired by Novo dvorsky's GSp(4)xGL(2) convolution) involving ametaplectic automorphic form on the double cover of GSp(4), the H-function on the double cover of GL(2), and a (non-metaplectic) Eisenstein series on GL(2), that yields a degree 5 Euler product. This Euler product is shown to have meromorphic continuation and a functional equation, and furthermore it has the same local Euler factors as the L-function of an automorphic form on GSp(4). As in [Shi, BH2], the Euler product found by Friedberg and Wong is explicitly constructed from the Whittaker coeEcients of the metaplectic automorphic form. Alternatively, Flicker [Fli], Kazhdan and Patterson [KaP2], and Flicker and Kazhdan [FliKa] have used the trace formula to generalize [Shi] by showing that (in many situations) there exists a correspondence between metaplectic automorphic forms and (non-metaplectic) automorphic forms. Indeed: Shimura actually proves in [Shi] that the Euler product (0.2) is the L-function of a holomorphic integral weight modular form. In using the trace formula, however: explicit information about the interplay between the metaplectic Fourier coefficients and the corresponding L-functions (see (0.2)) is not obtained. If a generalized Shimura correspondence does exist between generic metaplectic and non-metaplectic automorphic forms, then the associated Euler products obtained by Bump-Hofistein, Friedberg-Wong, and this paper will be the L-functions of the corresponding non-metaplectic forms. There is evidence that the degree 4 Euler product obtained in this paper is the L-function of an automorphic form on GSp(4) Savin [Sa] has shown that there is an algebra isomorphism between the local Iwahori Hecke algebra of GSp(4) and the local Iwahori Hecke algebra on the 3-fold cover of GSp(4) This suggests that; if a Shimura correspondent exists in this situation, it should be an automorphic form on GSp(4)e Since there is a representation of degree 4 on the L-group of GSp(4): automorphic forms on GSp(4) will have natural L-functions with Euler products of degree 4 In this sense, having a degree 4 Euler product is consistent with Savin's results. The main results of this paper will be found in Theorems 3.1 and 5.1, which are summarized as follows: Mairl Theorem. Suppose Q is a generic metaplectic C?lsp form of minimal level on the S-fold cover of G8p(4) that is an eigenf?lnction of all the lTecke operators. Then there is a degree J E?ller prod?lct, with a meromorphic contin?ltation, which cctn be explicitly constr?lcted from the Whittaker coefficients of f0Je This association is realized as a Shim?lra type Rankin-Selberg integral of 0 against an Eisenstein series ind?lced from a f)-f?lnction on the S-fold cover of GL(3). This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group W W and the Hecke algebra of double cosets of a parabolic subgroup of W . W. For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

This paper deals with a reducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) acts on the formal power series ring via ( 0.1 ) (0.1) . Then I ( f ) = ( ℓ i 1 ) ⊕ ( ℓ i 2 ) ⊕ ⋯ ⊕ ( ℓ i s ) I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}}) modulo some one dimensional s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representations where ( ℓ i ) (\ell _{i}) is an irreducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representation of dimension ℓ i \ell _{i} or empty set and { ℓ i 1 , ℓ i 2 , … , ℓ i s } ⊆ { ℓ 1 , ℓ 2 , … , ℓ r } \{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\} . Unlike classical invariant theory which deals only with irreducible action and 1–dimensional representations, we treat the reducible action and higher dimensional representations succesively.

In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character χ \chi to squarefree modulus q q , we prove the following upper bound: | ∑ 1 ⩽ n ⩽ N χ ( n ) | ⩽ c q log ⁡ q , \begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*} where c = 1 / ( 2 π 2 ) + o ( 1 ) c=1/(2\pi ^2)+o(1) for even characters and c = 1 / ( 4 π ) + o ( 1 ) c=1/(4\pi )+o(1) for odd characters, with an explicit o ( 1 ) o(1) term. This improves a result of Frolenkov and Soundararajan for large q q . We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of log ⁡ q \log {q} as in previous approaches and is an important factor for fully explicit bounds.

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in R d \mathbb {R}^d , killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

In this paper, we study the Dirichlet problem at infinity for harmonic maps between complete hyperbolic Hadamard surfaces. We will address the existence and uniqueness questions relating to the problem. In particular, we generalize results in the work of Li-Tam and Wan. 0. INTRODUCTION In a series of papers [L-T 1], [L-T 2] and [L-T 3], the first two authors considered the existence, uniqueness and boundary regularity theory for the Dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces. If we denote Em to be ther m-dimensional hyperbolic space with constant -1 sectional curvature, then using the Poincare' disk model, we can represent Hm by the unit m-disk D}m endowed with the Poincare metric ds2m = 4 ds2 where p denotes the Euclidean distance to the origin and ds2 is the Euclidean metric. The geometric compactification is given by the natural compactification of lEDm by the unit (m 1)-sphere Sm-1. Using this identification, a map from Sm-1 to S`1 can be viewed as a map from infinity of H"m to infinity of Hn. Let us summarize some of the results in [L-T 1], [L-T 2] and [L-T 3] as follows: Theorem (Li-Tam). Let q: S(m-1) -, (n-1) be a C1 map which has nowhere vanishing energy density. Then there exists a unique harmonic extension h Hm H Hn of X which is in CI(lrD"m, EDn"). Also, if X is in Ck,a(S(m-l), s(n-1)), for some 1 < k < m and some 0 < a, then h is in Ck,Y(lD"D, lEDn) for some O < y < a. We would like to mention that the existence part of this theorem for the special case when m = 2 = n and for C3 boundary maps which have nowhere vanishing energy density was independently proved by Akutagawa [A]. Using a different point of view, Wan [W] studied harmonic diffeomorphisms between hyperbolic 2-spaces via their associated Hopf differentials. By using the fact that [C-T] the Gauss map of a constant mean curvature cut in a Minkowski space-time is a harmonic map into hyperbolic space, Wan established that there is a one-to-one correspondence between the set of equivalent classes of constant mean curvature cuts and the equivalent classes of quadratic differentials in H2. Received by the editors December 12, 1994. 1991 Mathematics Subject Classification. Primary 58E20. The authors were partially supported by NSF grant #DMS9300422. c 1995 American Mathematical Society

In his famous paper on modular equations and approximations to π \pi , Ramanujan offers several series representations for 1 / π 1/\pi , which he claims are derived from "corresponding theories" in which the classical base q q is replaced by one of three other bases. The formulas for 1 / π 1/\pi were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.

This paper studies a minimax problem for functionals in Hilbert space in the form of G ( u ) = 1 2 ρ | | u | | 2 − g ( u ) G(u) = \frac {1} {2}\rho ||u|{|^2} - g(u) , where g ( u ) g(u) is Fréchet differentiable with weakly continuous derivative. If G G has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence u j ∈ H {u_j} \in H such that \[ g ′ ( u j ) = ρ j u j , ρ j → ρ , | | u j | | → ∞ , g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,||{u_j}|| \to \infty , \] and G ( u j ) G({u_j}) approximates the minimax value from above. If \[ γ ( t ) = sup | | u | | 2 = t g ( u ) \gamma (t) = \sup \limits _{||u|{|^2} = t} \;g(u) \] and \[ J 0 = ( 2 inf t 2 > t 1 > 0 γ ( t 2 ) − γ ( t 1 ) t 2 − t 1 , 2 sup t 2 > t 1 > 0 γ ( t 2 ) − γ ( t 1 ) t 2 − t 1 ) , {J_0} = \left ( {2\inf \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}},2\sup \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right ), \] then g ′ ( u ) = ρ u g\prime (u) = \rho u has a nonzero solution u u for a dense subset of ρ ∈ J 0 \rho \in {J_0} .

The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy’s inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1,8] to the values of the parameters which are inaccessible by the methods of these papers.

In this paper the dual Orlicz–Minkowski problem, a generalization of the L p L_p dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.

In this paper, we establish a large deviation principle for pairs of continuous and purely discontinuous additive functionals of symmetric Borel right processes on Lusin spaces. We also establish compact embedding results for the extended Dirichlet spaces of symmetric Markov processes that possess Green potential kernels.

Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure $\mu$. The {\it critical exponent} $\sigma^*$ is the supremum of the $\sigma$ such that $B^{\sigma}_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied., Comment: 34 pages,15 figures

In this paper, we study Lagrangian mean curvature flow (LMCF) of asymptotically conical (AC) Lagrangian submanifolds asymptotic to a union of special Lagrangian cones. Since these submanifolds are non-compact, we establish a short-time existence theorem for AC LMCF first. Then we focus on the equivariant, almost-calibrated case and prove long-time existence and convergence results. In particular, under certain smallness assumptions on the initial data, we show that the equivariant, almost-calibrated AC LMCF converges to a Lagrangian catenoid or an Anciaux’s expander.

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + \alpha D$ for $\alpha \in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.

We prove a Picard theorem for holomorphic maps from C to a quadric hypersurface. This implies a theorem on the number of directions in general position omitted by the normals to a minimal surface of the conformal type of C. The distribution of the normals to a two-dimensional minimal surface in Rn has been studied by Chern and Osserman [ 1]. This paper is concerned only with a special case of their theorem. For a minimal surface of the type of the entire plane C, Chern and Osserman prove that, if the normals to the surface omit n + 1 directions in general position, where n is the dimension of the ambient space, the image of the Gauss map lies in a proper linear subspace of CPn I . Theorem 1 of this paper improves on their result in two ways. First, it is only assumed that the normals omit n directions in general position. Secondly, we prove that the image of the Gauss map lies in a linear subspace of codimension two; in consequence, the minimal surface "decomposes" into a holomorphic function and a minimal surface in R -2 , in a sense that will be made precise below. The method, which derives from a paper of M. L. Green [4], is to apply value-distribution theory to maps into a quadric hypersurface instead of maps into projective space. In the definitions that follow we adopt the notation used by Hoffman and Osserman in their memoir [5], to which we refer for details. Let M be a Riemann surface and f: M Rn, where n > 2, be a nonconstant smooth map, with components (f1, ... , fn) . If z = x + iy is a local coordinate on M, let afk _ *fk k = X a The map f is called a minimal surface if the (Pk are holomorphic and satisfy the equation of conformality (1) + ..+ + 2 ? If the vector (P 1 ..P. ,k) is nonzero then it gives the homogeneous coordinates of some point in the complex projective space CPn-I . Since the (k are Received by the editors June 19, 1987. Presented to the Special Session on Differential Geometry, April 25, 1987, at the Society's 834th meeting in Newark, New Jersey (Abstract 834-53-28). 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10; Secondary 30D35. (?) 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

In this paper the study of a functional calculus for subnormal ntuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined. An n-tuple of commuting operators S = (S1, . . ., Sn) on a Hilbert space Z is subnormal if there is an n-tuple N = (Nl, ..., Nn) of commuting normal operators on a Hilbert space Z that contains Z such that for 1 < j < n, Nj}° C }° and Sj = Njlt. For any commuting n-tuple of operators S there is a notion of spectrum, the Taylor spectrum of S [30] (also see [12]), denoted by v(S). This spectrum is a nonempty compact subset of (Un. In this general theory it is possible to define +(S) for any function + analytic in a neighborhood of v(S). This functional calculus generalizes the usual Riesz functional calculus for a single operator. When S is assumed to be a subnormal n-tuple, however, a much richer functional calculus is possible. In this case 0(S) can be defined for functions 0 that are weak* limits of analytic functions and 0(S) becomes a subnormal operator. The central question in this development is "What are the properties of this operator 0(S) and what are the relations between the operator and the function X ?" More generally, if S is an n-tuple and Xl, . . ., Xq are analytic functions defined in a neighborhood of the Taylor spectrum of S, then 0(S)-(0l (S), .... Xq(S)) is a commuting q-tuple of operators. If S is subnormal, then 0(S) can be defined for X = (0l, ..., Xq) consisting of functions that are weak* limits Received by the editors January 31, 1989 and, in revised form, July 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B20, 47A60; Secondary 32A35, 47B38. The results here were delivered in a talk at the American Mathematical Society Summer Institute in Operator Theory held at the University of New Hampshire in July 1988. During the preparation of this paper partial support was furnished by National Science Foundation grant DMS 87-00835. (32)1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations. The purpose of this paper is to obtain much better results on the existence and uniqueness of strictly positive stationary (that is time-independent) solutions of