Long-range potential scattering by Enss's method in two Hilbert spaces
Denis A. W.
White
1-33
Abstract: Existence and completeness of wave operators is established by a straightforward transposition of the original short range result of Enss into an appropriate two-Hilbert space setting. Applied to long range quantum mechanical potential scattering, this result in conjunction with recent work of Isozaki and Kitada reduces the problem of proving existence and completeness of wave operators to that of approximating solutions of certain partial differential equations on cones in phase space. As an application existence and completeness of wave operators is established for Schrödinger operators with a long range multiplicative and possibly rapidly oscillating potential.
Brownian excursions from hyperplanes and smooth surfaces
Krzysztof
Burdzy
35-57
Abstract: A skew-product decomposition of the $n$-dimensional $ (n \geq 2)$ Brownian excursion law from a hyperplane is obtained. This is related to a Kolmogorov-type test for excursions from hyperplanes. Several results concerning existence, uniqueness and form of Brownian excursion laws from sufficiently "flat" surfaces are given. Some of these theorems are potential-theoretic in spirit. An extension of the results concerning excursion laws to an exit system in a Lipschitz domain is supplied.
Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves
R.
Biggers;
M.
Fried
59-70
Abstract: Let $ ({C_1}, \ldots ,{C_r}G) = ({\mathbf{C}},G)$ be an $r$-tuple consisting of a transitive subgroup $ G$ of ${S_m}$ and $r$ conjugacy classes ${C_1}, \ldots ,{C_r}$ of $G$. We consider the concept of the moduli space $ \mathcal{H}({\mathbf{C}},G)$ of compact Riemann surface covers of the Riemann sphere of Nielsen class $({\mathbf{C}},G)$. The irreducibility of $ \mathcal{H}({\mathbf{C}},G)$ is equivalent to the transitivity of a specific permutation representation of the Hurwitz monodromy group $(\S1)$, but there are few general tools to decide questions about this representation. Theorem 2 gives a class of examples of $({\mathbf{C}},G)$ for which $ \mathcal{H}({\mathbf{C}},G)$ is irreducible. As an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus $g$ curves (for which Deligne and Mumford [ ${\mathbf{DM}}$, Theorem 5.15] applied Teichmüller theory and Dehn's theorem). This contrasts with the examples of $ ({\mathbf{C}},G)$ in $[{\mathbf{BFr}}]$ for which $ \mathcal{H}({\mathbf{C}},G)$ is reducible. These kinds of questions combined with the study of the existence of rational subvarieties of $ \mathcal{H}({\mathbf{C}},G)$ have application to the realization of a group $ G$ as the Galois group of a regular extension of $ \mathbb{Q}(t)\;[{\mathbf{Fr3}},\S4]$.
Defining equations for real analytic real hypersurfaces in ${\bf C}\sp n$
John P.
D’Angelo
71-84
Abstract: A defining function for a real analytic real hypersurface can be uniquely written as $2\operatorname{Re} (H) + E$, where $H$ is holomorphic and $ E$ contains no pure terms. We study how $H$ and $E$ change when we perform a local biholomorphic change of coordinates, or multiply by a unit. One of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of $ E$ (expanded in terms of $ H$) beyond ${E_{00}}$ that serve as obstructions to writing a defining equation as $2\operatorname{Re} (h) + e$, where $e$ is independent of $ h$. We also find necessary pluriharmonic obstructions to doing this, which arise from the easier case of attempting to straighten the hypersurface.
Lyapunov exponents for a stochastic analogue of the geodesic flow
A. P.
Carverhill;
K. D.
Elworthy
85-105
Abstract: New invariants for a Riemannian manifold are defined as Lyapunov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature. For surfaces of constant negative curvature a direct method via the Doob $ h$-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.
The singularities of the $3$-secant curve associated to a space curve
Trygve
Johnsen
107-118
Abstract: Let $C$ be a curve in ${P^3}$ over an algebraically closed field of characteristic zero. We assume that $C$ is nonsingular and contains no plane component except possibly an irreducible conic. In [ $ {\mathbf{GP}}$] one defines closed $r$-secant varieties to $C$, $r \in N$. These varieties are embedded in $ G$, the Grassmannian of lines in ${P^3}$. Denote by $T$ the $3$-secant variety (curve), and assume that the set of $4$-secants is finite. Let $\tilde T$ be the curve obtained by blowing up the ideal of $4$-secants in $T$. The curve $\tilde T$ is in general not in $G$. We study the local geometry of $ \tilde T$ at any point whose fibre of the blowing-up map is reduced at the point. The multiplicity of $\tilde T$ at such a point is determined in terms of the local geometry of $C$ at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of $\tilde T$ at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.
Regions of variability for univalent functions
Peter
Duren;
Ayşenur
Ünal
119-126
Abstract: Let $S$ be the standard class of univalent functions in the unit disk, and let ${S_0}$ be the class of nonvanishing univalent functions $g$ with $g(0) = 1$. It is shown that the regions of variability $\{ g(r):g \in {S_0}\}$ and $\{ (1 - {r^2})f\prime(r):f \in S\}$ are very closely related but are not quite identical.
Weighted weak $(1,1)$ and weighted $L\sp p$ estimates for oscillating kernels
Sagun
Chanillo;
Douglas S.
Kurtz;
Gary
Sampson
127-145
Abstract: Weak type $ (1,1)$ and strong type $ (p,p)$ inequalities are proved for operators defined by oscillating kernels. The techniques are sufficiently general to derive versions of these inequalities using weighted norms.
Jones polynomials of alternating links
Kunio
Murasugi
147-174
Abstract: Let $ {J_K}(t) = {a_r}{t^r} + \cdots + {a_s}{t^s},r > s$, be the Jones polynomial of a knot $K$ in ${S^3}$. For an alternating knot, it is proved that $r - s$ is bounded by the number of double points in any alternating projection of $K$. This upper bound is attained by many alternating knots, including $2$-bridge knots, and therefore, for these knots, $r - s$ gives the minimum number of double points among all alternating projections of $K$. If $K$ is a special alternating knot, it is also proved that ${a_s} = 1$ and $s$ is equal to the genus of $K$. Similar results hold for links.
Affine manifolds and orbits of algebraic groups
William M.
Goldman;
Morris W.
Hirsch
175-198
Abstract: This paper is the sequel to The radiance obstruction and parallel forms on affine manifolds (Trans. Amer. Math. Soc. 286 (1984), 629-649) which introduced a new family of secondary characteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold ${M^n}$ to the action on ${{\mathbf{R}}^n}$ of the algebraic hull ${\mathbf{A}}(\Gamma )$ of the affine holonomy group $\Gamma \subseteq \operatorname{Aff}({{\mathbf{R}}^n})$. A main technical result of the paper is that if $M$ has a nonzero cohomology class represented by a parallel $k$-form, then every orbit of ${\mathbf{A}}(\Gamma )$ has dimension $\geq k$. When $M$ is compact, then ${\mathbf{A}}(\Gamma )$ acts transitively provided that $M$ is complete or has parallel volume; the converse holds when $\Gamma$ is nilpotent. A $4$-dimensional subgroup of $ \operatorname{Aff}({{\mathbf{R}}^3})$ is exhibited which does not contain the holonomy group of any compact affine $3$-manifold. When $M$ has solvable holonomy and is complete, then $M$ must have parallel volume. Conversely, if $ M$ has parallel volume and is of the homotopy type of a solvmanifold, then $ M$ is complete. If $ M$ is a compact homogeneous affine manifold or if $M$ possesses a rational Riemannian metric, then it is shown that the conditions of parallel volume and completeness are equivalent.
Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh
Rong Qing
Jia
199-212
Abstract: Let $\Delta$ be the mesh in the plane obtained from a uniform square mesh by drawing in the north-east diagonal in each square. Let $\pi _{k,\Delta }^\rho$ be the space of bivariate piecewise polynomial functions in ${C^\rho }$, of total degree $\leq k$, on the mesh $\Delta$. Let $ m(k,\rho )$ denote the approximation order of $\pi _{k,\Delta }^\rho$. In this paper, an upper bound for $m(k,\rho )$ is given. In the space $3 \leq 2k - 3\rho \leq 7$, the exact values of $m(k,\rho )$ are obtained: \begin{displaymath}\begin{array}{*{20}{c}} {m(k,\rho ) = 2k - 2\rho - 1} ... ...or}}\;2k - 3\rho = 5,6\;{\text{or}}\;7.} \end{array} \end{displaymath} In particular, this result answers negatively a conjecture of de Boor and Höllig.
Weighted and vector-valued inequalities for potential operators
Francisco J.
Ruiz Blasco;
José L.
Torrea Hernández
213-232
Abstract: In this paper we develop some aspect of a general theory parallel to the Calderón-Zygmund theory for operator valued kernels, where the operators considered map functions defined on ${R^n}$ into functions defined on $ R_ + ^{n + 1} = {R^n} \times [0,\infty )$. In particular, we apply the obtained results to get vector-valued inequalities for the Poisson integral and fractional integrals. Some weighted norm inequalities are also considered for fractional integrals.
$S\sp 1$-equivariant function spaces and characteristic classes
Benjamin M.
Mann;
Edward Y.
Miller;
Haynes R.
Miller
233-256
Abstract: We determine the structure of the homology of the Becker-Schultz space $SG({S^1}) \simeq Q({\mathbf{C}}P_ + ^\infty \wedge {S^1})$ of stable ${S^1}$-equivariant self-maps of spheres (with standard free ${S^1}$-action) as a Hopf algebra over the Dyer-Lashof algebra. We use this to compute the homology of $BSG({S^1})$. Along the way, we give a fresh account of the partially framed transfer construction and the Becker-Schultz homotopy equivalence. We compute the effect in homology of the "${S^1}$-transfers" ${\mathbf{C}}P_ + ^\infty \wedge {S^1} \to Q((B{{\mathbf{Z}}_{{p^n}}})_+ ),n \geq 0$, and of the equivariant $ J$-homomorphisms $ SO \to Q({\mathbf{R}}P_ + ^\infty )$ and $U \to Q({\mathbf{C}}P_ + ^\infty \wedge {S^1})$. By composing, we obtain $ U \to Q{S^0}$ in homology, answering a question of J. P. May.
Well-posedness of higher order abstract Cauchy problems
Frank
Neubrander
257-290
Abstract: The paper is concerned with differential equations of the type $\displaystyle {u^{(n + 1)}}(t) - A{u^{(n)}}(t) - {B_1}{u^{(n - 1)}}(t) - \cdots - {B_n}u(t) = 0$ ($\ast$) in a Banach space $E$ where $A$ is a linear operator with dense domain $ D(A)$ and ${B_1}, \ldots ,{B_n}$ are closed linear operators with $ D(A) \subset D({B_k})$ for $1 \leq k \leq n$. The main result is the equivalence of the following two statements: (a) $A$ has nonempty resolvent set and for every initial value $({x_0}, \ldots ,{x_n}) \in {(D(A))^{n + 1}}$ the equation $( \ast )$ has a unique solution in ${C^{n + 1}}({{\mathbf{R}}^ + },E) \cap {C^n}({{\mathbf{R}}^n},[D(A)])([D(A)]$ denotes the Banach space $ D(A)$ endowed with the graph norm); (b) $A$ is the generator of a strongly continuous semigroup. Under additional assumptions on the operators $ {B_k}$, which are frequently fulfilled in applications, we obtain continuous dependence of the solutions on the initial data; i.e., well-posedness of $( \ast )$. Using Laplace transform methods, we give explicit expressions for the solutions in terms of the operators $A$, ${B_k}$. The results are then used to discuss strongly damped semilinear second order equations.
The problem of embedding $S\sp n$ into ${\bf R}\sp {n+1}$ with prescribed Gauss curvature and its solution by variational methods
V. I.
Oliker
291-303
Abstract: A way to recover a closed convex hypersurface from its Gauss curvature is to find a positive function over ${S^n}$ whose graph would represent the hypersurface in question. Then one is led to a nonlinear elliptic problem of Monge-Ampère type on $ {S^n}$. Usually, geometric problems involving operators of this type are too complicated to be suggestive for a natural functional whose critical points are candidates for solutions of such problems. It turns out that for the problem indicated in the title, such a functional exists and has interesting geometric properties. With the use of this functional, we obtain new existence results for hypersurfaces with prescribed curvature as well as strengthen some that are already known.
Eigenvalues of elliptic boundary value problems with an indefinite weight function
Jacqueline
Fleckinger;
Michel L.
Lapidus
305-324
Abstract: We consider general selfadjoint elliptic eigenvalue problems (P) $\displaystyle \mathcal{A}u = \lambda r(x)u,$ in an open set $\Omega \subset {{\mathbf{R}}^k}$. Here, the operator $ \mathcal{A}$ is positive and of order $2m$ and the "weight" $r$ is a function which changes sign in $ \Omega$ and is allowed to be discontinuous. A scalar $\lambda$ is said to be an eigenvalue of $({\text{P}})$ if $\mathcal{A}u = \lambda ru$--in the variational sense--for some nonzero $u$ satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of $ ({\text{P}})$, under suitable assumptions. In the case when $\Omega$ is bounded, we assumed Dirichlet or Neumann boundary conditions. When $ \Omega$ is unbounded, we work with operators of "Schrödinger type"; if we set $ r \pm = \max ( \pm r,0)$, two cases appear naturally: First, if $\Omega$ is of "weighted finite measure" (i.e., $ \int_\Omega {{{({r_ + })}^{k/2m}} < + \infty \;} {\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} < + \infty }$), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if $\Omega$ is of "weighted infinite measure" (i.e., $\int_\Omega {{{({r_ + })}^{k/2m}} = + \infty \;{\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} = + \infty } }$), our results extend the de Wet-Mandl formula (which is classical for Schrödinger operators with weight $r \equiv 1$). When $\Omega$ is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian.
Random recursive constructions: asymptotic geometric and topological properties
R. Daniel
Mauldin;
S. C.
Williams
325-346
Abstract: We study some notions of "random recursive constructions" in Euclidean $ m$-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiński curve or Menger curve. We demonstrate that associated with each such construction is a "universal" number $\alpha$ such that almost surely the random object has Hausdorff dimension $\alpha$. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula.
Embedding strictly pseudoconvex domains into balls
Franc
Forstnerič
347-368
Abstract: Every relatively compact strictly pseudoconvex domain $D$ with $ {{\mathbf{C}}^2}$ boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each $n \geq 2$ there exist bounded strictly pseudoconvex domains $D$ in $ {\mathbb{C}^n}$ with real-analytic boundary such that no proper holomorphic map from $D$ into any finite dimensional ball extends smoothly to $ \overline D$.
Splitting strongly almost disjoint families
A.
Hajnal;
I.
Juhász;
S.
Shelah
369-387
Abstract: We say that a family $\mathcal{A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $\vert A \cap B\vert < \kappa$, e.g. that $ \vert A \cap B\vert < \sigma < \kappa$, is assumed for $A$, $B \in \mathcal{A}$. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each $A \in \mathcal{A}$ there is $ F(A) \in {[A]^{ < \kappa }}$ so that $\{ A\backslash F(A):A \in \mathcal{A}\}$ is disjoint. On the other hand, we get from a supercompact cardinal the consistency of $ {\text{GCH}}$ plus the existence of a family $\mathcal{A} \subset {[{\omega _{\omega + 1}}]^{{\omega _1}}}$ whose elements have pairwise finite intersections and such that it does not even have property $B$. This solves an old problem raised in [4]. The same example is also used to produce a graph of chromatic number $ {\omega _2}$ on ${\omega _{\omega + 1}}$ that does not contain $[\omega ,\omega ]$, answering a problem from [5]. We also have applications of our results to "splitting" certain families of closed subsets of a topological space. These improve results from [ $ {\mathbf{3}},{\mathbf{12}}$ and $ {\mathbf{13}}$].
Uniqueness results for homeomorphism groups
Robert R.
Kallman
389-396
Abstract: Let $X$ be a separable metric manifold and let $ \mathcal{H}(X)$ be the homeomorphism group of $X$. Then $ \mathcal{H}(X)$ has a unique topology in which it is a complete separable metric group. Similar results are demonstrated for a much wider class of spaces, $X$, and for many subgroups of the homeomorphism group.
On the generic structure of cohomology modules for semisimple algebraic groups
Henning Haahr
Andersen
397-415
Abstract: Let $G$ be a connected semisimple algebraic group over a field of positive characteristic. Denote by $B$ a Borel subgroup. Our main result says that generically the cohomology modules for line bundles on $ G/B$ have simple socles and simple heads, and we identify the corresponding highest weights. As one of the consequences we discover a certain symmetry among extensions of simple modules for $G$.
The associative forms of the graded Cartan type Lie algebras
Rolf
Farnsteiner
417-427
Abstract: This paper determines the Cartan type Lie algebras that possess a nonsingular associative form.
Erratum to: ``Generic algebras'' [Trans. Amer. Math. Soc. {\bf 275} (1983), no. 2, 497--510; MR0682715 (84h:18010)]
John
Isbell
429