Calibrations on ${\bf R}\sp 8$
J.
Dadok;
R.
Harvey;
F.
Morgan
1-40
Abstract: Calibrations are a powerful tool for constructing minimal surfaces. In this paper we are concerned with $ 4$-manifolds $M \subset {{\mathbf{R}}^8}$. If a differential form $\varphi \in { \bigwedge ^4}{{\mathbf{R}}^8}$ calibrates all tangent planes of $M$ then $M$ is area minimizing. For $\varphi$ in one of several large subspaces of $ { \bigwedge ^4}{{\mathbf{R}}^8}$ we compute its comass, that is the maximal value of $\varphi$ on the Grassmannian of oriented $ 4$-planes. We then determine the set $G(\varphi ) \subset G(4,\,{{\mathbf{R}}^8})$ on which this maximum is attained. These are the planes calibrated by $\varphi$, the possible tangent planes to a manifold calibrated by $\varphi$. The families of calibrations obtained include the well-known examples: special Lagrangian, Kähler, and Cayley.
Curves of genus $2$ with split Jacobian
Robert M.
Kuhn
41-49
Abstract: We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If $ X$ is a curve of genus $ 2$, and $f:X \to E$ a map from $X$ to an elliptic curve, then $ X$ has split jacobian. It is not true that a complement to $E$ in the jacobian of $X$ is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve $E'$ and algebraic
Sieving the positive integers by small primes
D. A.
Goldston;
Kevin S.
McCurley
51-62
Abstract: Let $Q$ be a set of primes that has relative density $\delta$ among the primes, and let $\phi (x,\,y,\,Q)$ be the number of positive integers $\leqslant x$ that have no prime factor $\leqslant y$ from the set $Q$. Standard sieve methods do not seem to give an asymptotic formula for $\phi (x,\,y,\,Q)$ in the case that $\tfrac{1}{2} \leqslant \delta < 1$. We use a method of Hildebrand to prove that $\displaystyle \phi (x,y,Q)\tilde{x}f(u)\prod\limits_{\mathop {p < y}\limits_{p \in Q} } {\left( {1 - \frac{1}{p}} \right)} $ as $x \to \infty$, where $u = \frac{{\log x}}{{\log y}}$ and $f(u)$ is defined by $\displaystyle {u^\delta }f(u) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{e^{{\... ...1 + t)}^{\delta - 1}}\;dt,} } & {u > 1.} \end{array} } \right.$ This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where $ Q$ consisted of all primes.
Construction of manifolds of positive scalar curvature
Rodney
Carr
63-74
Abstract: We prove that a regular neighborhood of a codimension $\geqslant 3$ subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact $4$-manifold with such a metric.
First- and second-order epi-differentiability in nonlinear programming
R. T.
Rockafellar
75-108
Abstract: Problems are considered in which an objective function expressible as a max of finitely many ${C^2}$ functions, or more generally as the composition of a piecewise linear-quadratic function with a ${C^2}$ mapping, is minimized subject to finitely many ${C^2}$ constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical first-and second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.
Exact bounds for the stochastic upward matching problem
WanSoo T.
Rhee;
Michel
Talagrand
109-125
Abstract: We draw at random independently and according to the uniform distribution two sets of $n$ points of the unit square. We consider a maximum matching of points of the first set with points of the second set with the restriction that a point can be matched only with a point located at its upper right. Then with probability close to one, the number of unmatched points is of order ${n^{1/2}}{(\log n)^{3/4}}$.
Characteristic multipliers and stability of symmetric periodic solutions of $\dot x(t)=g(x(t-1))$
Shui-Nee
Chow;
Hans-Otto
Walther
127-142
Abstract: We study the scalar delay differential equation $\dot x(t) = g(x(t - 1))$ with negative feedback. We assume that the nonlinear function $g$ is odd and monotone. We prove that periodic solutions $x(t)$ of slowly oscillating type satisfying the symmetry condition $x(t) = - x(t - 2)$, $t \in {\mathbf{R}}$, are nondegenerate and have all nontrivial Floquet multipliers strictly inside the unit circle. This says that the periodic orbit $ \{ {x_t}:t \in {\mathbf{R}}\}$ in the phase space $ C[ - 1,\,0]$ is orbitally exponentially asymptotically stable.
Functions that preserve the uniform distribution of sequences
William
Bosch
143-152
Abstract: In this paper, necessary and sufficient conditions are given for certain functions to preserve the uniform distribution of sequences. An analytic condition allows the construction of examples. An application is also given.
A strong contractivity property for semigroups generated by differential operators
Robert M.
Kauffman
153-169
Abstract: Frequently, nonconservative semigroups generated by partial differential operators in $ {L_{2,\rho }}({R^k})$ have the property that initial conditions which are large at $\vert x\vert = \infty$ become immediately small at infinity for all $t > 0$. This property is related to the rate of decay of eigenfunctions of the differential operator. In this paper this phenomenon is investigated for a large class of differential operators of second and higher order. New estimates on the rate of decay of the eigenfunctions are included, which are related in special cases to those of Agmon.
Conditional gauge and potential theory for the Schr\"odinger operator
M.
Cranston;
E.
Fabes;
Z.
Zhao
171-194
Abstract: This paper extends the Conditional Gauge Theorem to more general operators and less regular domains than in previous works. We use this to obtain potential-theoretic results for the Schrödinger equation.
The connection map for attractor-repeller pairs
Christopher
McCord
195-203
Abstract: In the Conley index theory, the connection map of the homology attractor-repeller sequence provides a means of detecting connecting orbits between a repeller and attractor in an isolated invariant set. In this work, the connection map is shown to be additive: under suitable decompositions of the connecting orbit set, the connection map of the invariant set equals the sum of the connection maps of the decomposition elements. This refines the information provided by the homology attractor-repeller sequence. In particular, the properties of the connection map lead to a characterization of isolated invariant sets with hyperbolic critical points as an attractor-repeller pair.
Multiple fibers on rational elliptic surfaces
Brian
Harbourne;
William E.
Lang
205-223
Abstract: Our main result, Theorem (0.1), classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic. One result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the Kodaira fiber-types for which no examples were previously known.
Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II
M. S.
Berger;
P. T.
Church;
J. G.
Timourian
225-244
Abstract: Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem $\displaystyle \Delta u + \lambda u - {u^3} = g\quad {\text{on}}\;\Omega ,\qquad u\vert\partial \Omega = 0,$ where $\Omega \subset {{\mathbf{R}}^n}(n \leqslant 4)$ is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.
Isometry groups of Riemannian solvmanifolds
Carolyn S.
Gordon;
Edward N.
Wilson
245-269
Abstract: A simply connected solvable Lie group $R$ together with a left-invariant Riemannian metric $g$ is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds $(R,\,g)$ and $ (R' ,\,g' )$ may be isometric even when $R$ and $R'$ are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization $ (R,\,g)$ of a given solvmanifold, (ii) describing the embedding of $R$ in the full isometry group $ I(R,\,g)$, and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of $I(R,\,g)$ for arbitrary solvmanifolds $(R,\,g)$.
Regularity of solutions of two-dimensional Monge-Amp\`ere equations
Friedmar
Schulz;
Liang Yuan
Liao
271-277
Abstract: In the paper we investigate the regularity of solutions $z(x,\,y) \in {C^{1,1}}(\Omega )$, resp. ${C^{1,1}}(\overline \Omega )$ of elliptic Monge-Ampére equations of the form $\displaystyle Ar + 2Bs + Ct + (rt - {s^2}) = E.$ It is shown that $z(x,\,y) \in {C^{2,\alpha }}(\Omega )$, resp. $ {C^{2,\alpha }}(\overline \Omega )$, with corresponding a priori estimates, if $ A,\,B,\,C,\,E \in {C^\alpha }(\Omega \times {{\mathbf{R}}^3})$. The results are deduced via the Campanato technique for equations of variational structure invoking a Legendre-like transformation.
Some sharp inequalities for martingale transforms
K. P.
Choi
279-300
Abstract: Two sharp inequalities for martingale transforms are proved. These results extend some earlier work of Burkholder. The inequalities are then extended to stochastic integrals and differentially subordinate martingales.
Spinor bundles on quadrics
Giorgio
Ottaviani
301-316
Abstract: We define some stable vector bundles on the complex quadric hypersurface $ {Q_n}$ of dimension $ n$ as the natural generalization of the universal bundle and the dual of the quotient bundle on ${Q_4} \simeq \operatorname{Gr} (1,\,3)$. We call them spinor bundles. When $n = 2k - 1$ there is one spinor bundle of rank ${2^{k - 1}}$. When $n = 2k$ there are two spinor bundles of rank ${2^{k - 1}}$. Their behavior is slightly different according as $ n \equiv 0\;(\bmod 4)$ or $ n \equiv 2\;(\bmod 4)$. As an application, we describe some moduli spaces of rank $ 3$ vector bundles on $ {Q_5}$ and ${Q_6}$.
Constant isotropic submanifolds with $4$-planar geodesics
Jin Suk
Pak;
Kunio
Sakamoto
317-333
Abstract: Let $f$ be an isometric immersion of a Riemannian manifold $M$ into $ \overline M$. We prove that if $f$ is constant isotropic, $4$-planar geodesic and $\overline M$ is a Euclidean sphere, then $ M$ is isometric to one of compact symmetric spaces of rank equal to one and $ f$ is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic, $4$-planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature.
Pseudodifferential operators with coefficients in Sobolev spaces
Jürgen
Marschall
335-361
Abstract: Pseudo-differential operators with coefficients in Sobolev spaces $ {H^{r,q}},1 \leqslant q \leqslant \infty$, and their adjoints are studied on Hardy-Sobolev spaces ${H^{s,p}},\;0 < p \leqslant \infty$. A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved.
Harmonically immersed surfaces of ${\bf R}\sp n$
Gary R.
Jensen;
Marco
Rigoli
363-372
Abstract: Some generalizations of classical results in the theory of minimal surfaces $ f:M \to {{\mathbf{R}}^n}$ are shown to hold in the more general case of harmonically immersed surfaces.
Bonnesen-style inequalities for Minkowski relative geometry
J. R.
Sangwine-Yager
373-382
Abstract: Two Bonnesen-style inequalities are obtained for the relative inradius of one convex body with respect to another in $ n$-dimensional space. Both reduce to the known planar inequality; one sharpens the relative isoperimetric inequality, the other states that a quadratic polynomial is negative at the inradius. Circumradius inequalities follow.
Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are locally connected
Janusz R.
Prajs
383-394
Abstract: We prove that each homogeneous continuum which topologically contains an $ n$-dimensional unit cube and lies in $(n + 1)$-dimensional Euclidean space is locally connected.
Embedding graphs into colored graphs
A.
Hajnal;
P.
Komjáth
395-409
Abstract: If $X$ is a graph, $\kappa$ a cardinal, then there is a graph $ Y$ such that if the vertex set of $Y$ is $\kappa$-colored, then there exists a monocolored induced copy of $X$; moreover, if $X$ does not contain a complete graph on $ \alpha$ vertices, neither does $Y$. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph $X$ such that for every graph $Y$ there is a two-coloring of the edges of $ Y$ such that there is no monocolored induced copy of $X$. Similarly, a triangle-free $X$ may exist such that every $ Y$ must contain an infinite complete graph, assuming that coloring $Y$'s edges with countably many colors a monocolored copy of $X$ always exists.
Coordinatization in superstable theories. II
Steven
Buechler
411-417
Abstract: In this paper we prove Theorem A. Suppose that $ T$ is superstable and $ U(a/A) = \alpha + 1$, for some $\alpha$. Then in ${T^{{\text{eq}}}}$ there is a $c \in \operatorname{acl} (Aa)\backslash \operatorname{acl} (A)$ such that one of the following holds. (i) $ U(c/A) = 1$. (ii) $ \operatorname{stp} (c/A)$ has finite Morley rank. In fact, this strong type is semiminimal with respect to a strongly minimal set.
Area-minimizing integral currents with boundaries invariant under polar actions
Julian C.
Lander
419-429
Abstract: Let $G$ be a compact, connected subgroup of $SO(n)$ acting on $ {{\mathbf{R}}^n}$, and let the action of $G$ be polar. (Polar actions include the adjoint action of a Lie group $H$ on the tangent space to the symmetric space $ G/H$ at the identity coset.) Let $B$ be an $(m - 1)$-dimensional submanifold without boundary, invariant under the action of $G$, and lying in the union of the principal orbits of $G$. It is shown that, if $S$ is an area-minimizing integral current with boundary $B$, then $S$ is invariant under the action of $G$. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.
Errata to: ``Nonsingular quadratic differential equations in the plane'' [Trans. Amer. Math. Soc. {\bf 301} (1987), no. 2, 845--859; MR0882718 (88b:58109)]
M. I. T.
Camacho;
C. F. B.
Palmeira
431