A selection theorem for topological convex structures
M.
van de Vel
463-496
Abstract: A continuous selection theorem has been obtained for multivalued functions, the values of which are convex sets of certain synthetic convex structures. Applications are given related with superextensions, (semi)lattices, spaces of order arcs, trees, Whitney levels in hyperspaces, and geometric topology. Applications to traditional convexity in vector spaces involve Beer's approximation theorem and a fixed point theorem of Dugundji-Granas. Some other applications (a.o. an invariant arc theorem) appear elsewhere.
Attracting and repelling point pairs for vector fields on manifolds. I
Gabriele
Meyer
497-507
Abstract: Consider a compact, connected, $n$-dimensional, triangulable manifold $ M$ without boundary, embedded in $ {{\mathbf{R}}^{n + 1}}$ and a continuous vector field on $M$, given as a map $f$ from $M$ to ${S^n}$ of degree not equal to 0 or ${( - 1)^{n + 1}}$. In this paper it is shown that there exists at least one pair of points $ x$, $y \in M$ satisfying both $f(x) = - f(y)$ and $f(x) = \frac{{x - y}} {{\vert\vert x - y\vert\vert}}$. Geometrically, this means, that the points and the vectors lie on one straight line and the vector field is "repelling". Similarly, if the degree of $ f$ is not equal to 0 or $ 1$, then there exists at least one "attracting" pair of points $x$, $y \in M$ satisfying both $f(x) = - f(y)$ and $f(x) = \frac{{y - x}} {{\vert\vert y - x\vert\vert}}$. The total multiplicities are $ \frac{{k \bullet (k + {{( - 1)}^n})}} {2}$ for repelling pairs and $\frac{{k \bullet (k - 1)}} {2}$ for attracting pairs. In the proof, we work with close simplicial approximations of the map $f$, using Simplicial, Singular and Čech Homology Theory, Künneth's Theorem, Hopf's Classification Theorem and the algebraic intersection number between two $n$-dimensional homology cycles in a $ 2n$-dimensional space. In the case of repelling pairs, we intersect the graph of $ f$ in $M \times {S^n}$ with the set of points $(x,\frac{{x - y}} {{\vert\vert x - y\vert\vert}}) \in M \times {S^n}$, where $x$ and $y$ satisfy that $ f(x) = - f(y)$. In order to show that this set carries the homology $ (k,k) \in {H_n}(M \times {S^n},{\mathbf{Z}})$, we study the set ${A_f} \equiv \{ (x,y) \in M \times M\vert f(x) = - f(y)\}$ in a simplicial setting. Let ${f_j}$ be a close simplicial approximation of $f$. It can be shown, that ${A_{{f_j}}}$ is a homology cycle of dimension $ n$ with a natural triangulation and a natural orientation and that $ {A_f}$ and ${A_f}_j$ carry the same homology.
The product of two normal initially $\kappa$-compact spaces
Eric K.
van Douwen
509-521
Abstract: We prove that it is independent from $ {\text{ZFC}}$ that for every cardinal $\kappa$ the following statements are equivalent: (a) $\kappa$ is singular; (b) initial $ \kappa$-compactness (defined above the introduction) is productive; (c) initial $\kappa$-compactness is finitely productive; and (d) the product of two initially $\kappa $-compact normal spaces is initially $\kappa$-compact. In particular, MA$$ implies that there are two countably compact normal spaces whose product is not countably compact.
Semilinear evolution equations in Banach spaces with application to parabolic partial differential equations
Samuel M.
Rankin
523-535
Abstract: A theory for a class of semilinear evolution equations in Banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data.
Higher order commutators for vector-valued Calder\'on-Zygmund operators
Carlos
Segovia;
José L.
Torrea
537-556
Abstract: Weighted norm estimates for higher order commutators are obtained. The proof, that remain valid in the vector-valued case, are obtained as an application of some extrapolation results. The vector-valued version of the commutator theorem is applied to the Carleson operator, U.M.D. Banach spaces, approximate identities and maximal operators.
Cohen-Macaulayness of semi-invariants for tori
Michel
Van den Bergh
557-580
Abstract: In this paper we give a new method, in terms of one-parameter subgroups, to study semi-invariants for algebraic tori. In some cases we obtain extensions to results in [7]. In other cases we obtain different proofs.
Failure of cancellation for direct sums of line bundles
Richard G.
Swan
581-605
Abstract: In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules $L$ of rank $1$ over a commutative ring $R$ such that $L \oplus {L^{ - 1}}$ is stably free but not free. Examples are also given of projective modules for which the determinant map det: $\operatorname{Aut}(P) \to {R^{\ast} }$ is not onto. Some related topological questions are also discussed.
The value semigroups of prime divisors of the second kind in $2$-dimensional regular local rings
Sunsook
Noh
607-619
Abstract: In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a $2$-dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a $ 2$-dimensional regular local ring to have the same value semigroup is obtained.
Mountain impasse theorem and spectrum of semilinear elliptic problems
Kyril
Tintarev
621-629
Abstract: This paper studies a minimax problem for functionals in Hilbert space in the form of $G(u) = \frac{1} {2}\rho \vert\vert u\vert{\vert^2} - g(u)$, where $g(u)$ is Fréchet differentiable with weakly continuous derivative. If $G$ 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 ${u_j} \in H$ such that $\displaystyle g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,\vert\vert{u_j}\vert\vert \to \infty ,$ and $G({u_j})$ approximates the minimax value from above. If $\displaystyle \gamma (t) = \mathop {\sup }\limits_{\vert\vert u\vert{\vert^2} = t} \;g(u)$ and $\displaystyle {J_0} = \left( {2\mathop {\inf }\limits_{{t_2} > {t_1} > 0} \frac... ...{t_1} > 0} \frac{{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right),$ then $ g\prime (u) = \rho u$ has a nonzero solution $u$ for a dense subset of $\rho \in {J_0}$.
On a conjecture of Lin-Ni for a semilinear Neumann problem
Adimurthi;
S. L.
Yadava
631-637
Abstract: Let $\Omega$ be a bounded domain in ${\mathbb{R}^n}$ $ (n \geq 3)$ and $\lambda > 0$. We consider \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u + \lambda u = {u^{(n + 2... ... }} = 0} & {{\text{on}}} \; {\partial \Omega ,} \end{array} \end{displaymath} and show that for $\lambda$ sufficiently small, the minimal energy solutions are only constants.
A unique continuation property on the boundary for solutions of elliptic equations
Zhi Ren
Jin
639-653
Abstract: We prove the following conclusion: if $u$ is a harmonic function on a smooth domain $ \Omega$ in ${R^n}$ , $n \geq 3$ , or a solution of a general second-order linear elliptic equation on a domain $\Omega$ in ${R^2}$, and if there are ${x_0} \in \partial \Omega $ and constants $ a$, $b > 0$ such that $\vert u(x)\vert \leq a\exp \{ - b/\vert x - {x_0}\vert\}$ for $ x \in \Omega$, $\vert x - {x_0}\vert$ small, then $u = 0$ in $\Omega$ . The decay rate in our results is best possible by the example that $u =$ real part of $\exp \{ - 1/{z^\alpha }\} $ , $0 < \alpha < 1$ , is harmonic but not identically zero in the right complex half-plane.
An indirect method in the calculus of variations
F. H.
Clarke
655-673
Abstract: This article presents a new approach to the issue of the existence of solutions to the basic problem in the calculus of variations. The method is indirect, and applies to certain classes of of problems with slow or no growth, in addition to those satisfying the traditional coercivity condition. The proof hinges upon showing with the help of nonsmooth analysis that a certain value function is constant. Examples are given to illustrate the applicability of the results and the necessity of the hypotheses.
Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains
Norberto
Salinas
675-699
Abstract: The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the $ {C^{\ast} }$-algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain $\Omega$ appears thus naturally, and a decomposition series of the corresponding ${C^{\ast} }$-algebra is presented in terms of groupoid ${C^{\ast} }$-algebras associated with various parts of the boundary of the domain $ \Omega$.
Weak solutions of the porous medium equation in a cylinder
Björn E. J.
Dahlberg;
Carlos E.
Kenig
701-709
Abstract: We show that if $D \subset {{\mathbf{R}}^n}$ is a bounded domain with smooth boundary, and $u \in {L^m}(D \times (\varepsilon ,T))$, $u \geq 0$, solves $\frac{{\partial u}} {{\partial t}} = \Delta {u^m}$, $m > 1$, in the sense of distributions on $D \times (0,T)$, and vanishes on $\partial D \times (0,T)$ in a suitable weak sense, then $u$ is Hölder continuous in $\overline D \times (0,T)$.
Weak solutions of the porous medium equation
Björn E. J.
Dahlberg;
Carlos E.
Kenig
711-725
Abstract: We show that if $u \geq 0$, $u \in L_{{\text{loc}}}^m(\Omega )$, $\Omega \subset {{\mathbf{R}}^{n + 1}}$ solves $\partial u/\partial t = \Delta {u^m}$, $m > 1$ , in the sense of distributions, then $u$ is locally Hölder continuous in $\Omega$.
Uniqueness in bounded moment problems
Hans G.
Kellerer
727-757
Abstract: Let $(X,\mathfrak{A},\mu )$ be a $\sigma$-finite measure space and $\mathcal{K}$ be a linear subspace of ${\mathcal{L}_1}(\mu )$ with $\mathcal{K} = X$. The following inverse problem is treated: Which sets $A \in \mathfrak{A}$ are " $ \mathcal{K}$-determined" within the class of all functions $g \in {\mathcal{L}_\infty }(\mu )$ satisfying $ 0 \leq g \leq 1$ , i.e. when is $g = {1_A}$ the unique solution of $ \smallint fg\;d\mu = \smallint f{1_A}\;d\mu$, $f \in \mathcal{K}?$ Recent results of Fishburn et al. and Kemperman show that the condition $A = \{ f \geq 0\}$ for some $f \in \mathcal{K}$ is sufficient but not necessary for uniqueness. To obtain a complete characterization of all $ \mathcal{K}$-determined sets, $\mathcal{K}$ has to be enlarged to some hull $ {\mathcal{K}^{\ast} }$ by extending the usual weak convergence to limits not in $ {\mathcal{L}_1}(\mu )$. Then one of the main results states that $A$ is $ \mathcal{K}$-determined if and only if there is a representation $A = \{ {f^{\ast} } > 0\}$ and $X\backslash A = \{ {f^{\ast} } < 0\}$ for some ${f^{\ast}} \in {\mathcal{K}^{\ast} }$ .
Branched surfaces and attractors. I. Dynamic branched surfaces
Joe
Christy
759-784
Abstract: We show how, using ideas of R. F. Williams about branched surfaces, hyperbolic attractors of flows on three manifolds may be classified up to topological equivalence on an isolating neighborhood by a finite combinatorial object, a swaddled graph.
On injectivity in locally presentable categories
Jiří
Adámek;
Jiří
Rosický
785-804
Abstract: Classes of objects injective w.r.t. specified morphisms are known to be closed under products and retracts. We prove the converse: a class of objects in a locally presentable category is an injectivity class iff it is closed under products and retracts. This result requires a certain large-cardinal principle. We characterize classes of objects injective w.r.t. a small collection of morphisms: they are precisely the accessible subcategories closed under products and $\kappa$-filtered colimits. Assuming the (large-cardinal) Vopênka's principle, the accessibility can be left out. As a corollary, we solve a problem of ${\text{L}}$. Fuchs concerning injectivity classes of abelian groups. Finally, we introduce a weak concept of reflectivity, called cone reflectivity, and we prove that under Vopênka's principle all subcategories of locally presentable categories are cone reflective. Several open questions are formulated, e.g., does each topological space have a largest (non-${T_2}$) compactification?
Supercuspidal representations and the theta correspondence. II. ${\rm SL}(2)$ and the anisotropic ${\rm O}(3)$
David
Manderscheid
805-816
Abstract: A parametrization is given of the local theta correspondence attached to the reductive dual pair $ ({\text{SL}}_2(F),\;{\text{O}}(F))$ where $F$ is a nonarchimedean local field of odd residual characteristic and $ {\text{O}}$ is the orthogonal group of a ternary quadratic form which is anisotropic over $F$. The parametrization is in terms of inducing data. Various lattice models of the oscillator representation are used.
On Lagrange interpolation at disturbed roots of unity
Charles K.
Chui;
Xie Chang
Shen;
Le Fan
Zhong
817-830
Abstract: Let ${z_{nk}} = {e^{i{t_{nk}}}}$, $0 \leq {t_{n0}} < \cdots < {t_{nn}} < 2\pi$, $f$ a function in the disc algebra $A$, and $ {\mu _n} = \max \{ \vert{t_{nk}} - 2k\pi /(n + 1)\vert:0 \leq k \leq n\}$. Denote by ${L_n}(f;\; \cdot )$ the polynomial of degree $n$ that agrees with $f$ at $\{ {z_{nk}}:k = 0, \ldots ,n\}$ . In this paper, we prove that for every $p$, $0 < p < \infty$, there exists a ${\delta _p} > 0$, such that $\vert\vert{L_n}(f;\cdot) - f\vert{\vert _p} = O(\omega (f;\frac{1} {n}))$ whenever ${\mu _n} \leq {\delta _p}/(n + 1)$. It must be emphasized that $ {\delta _p}$ necessarily depends on $p$, in the sense that there exists a family $ \{ {z_{nk}}:k = 0, \ldots ,n\}$ with ${\mu _n} = {\delta _2}/(n + 1)$ and such that $\vert\vert{L_n}(f;\cdot) - f\vert{\vert _2} = O(\omega (f;\frac{1} {n}))$ for all $f \in A$, but $\sup \{ \vert\vert{L_n}(f;\cdot)\vert{\vert _p}:f \in A,\vert\vert f\vert{\vert _\infty } = 1\}$ diverges for sufficiently large values of $p$. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for $\{ {z_{nk}}\}$.
Remarks on prescribing Gauss curvature
Xingwang
Xu;
Paul C.
Yang
831-840
Abstract: We study the nonlinear partial differential equation for the problem of prescribing Gauss curvature $K$ on ${S^2}$ . We give an example of a rotationally symmetric $K$ for which the Kazdan-Warner obstruction is satisfied but the equation has no rotationally symmetric solution. On the other hand, we give a simple sufficient condition for solvability of the equation when $K$ is rotationally symmetric. Finally we give a sufficient condition for solvability when $ K$ is not necessarily rotationally symmetric.
$W\sp {2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients
Filippo
Chiarenza;
Michele
Frasca;
Placido
Longo
841-853
Abstract: We prove a well-posedness result in the class ${W^{2,p}} \cap W_0^{1,p}$ for the Dirichlet problem $\displaystyle \left\{ {\begin{array}{*{20}{c}} {Lu = f} & {{\text{a.e.}}\;{\tex... ...Omega }, {u = 0} & {{\text{on}}\;\partial \Omega }. \end{array} } \right.$ We assume the coefficients of the elliptic nondivergence form equation that we study are in ${\text{VMO}} \cap {L^\infty }$ .
Unlinking via simultaneous crossing changes
Martin
Scharlemann
855-868
Abstract: Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers.
Weighted norm inequalities for homogeneous singular integrals
Javier
Duoandikoetxea
869-880
Abstract: We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral.
Functional equations satisfied by intertwining operators of reductive groups
Chen-bo
Zhu
881-899
Abstract: This paper generalizes a recent work of Vogan and Wallach [VW] in which they derived a difference equation satisfied by intertwining operators of reductive groups. We show that, associated with each irreducible finite-dimensional representation, there is a functional equation relating intertwining operators. In this way, we obtain natural relations between intertwining operators for different series of induced representations.
Groups of dualities
Georgi D.
Dimov;
Walter
Tholen
901-913
Abstract: For arbitrary categories $ \mathcal{A}$ and $\mathcal{B}$ , the "set" of isomorphism-classes of dualities between $ \mathcal{A}$ and $\mathcal{B}$ carries a natural group structure. In case $\mathcal{A}$ and $\mathcal{B}$ admit faithful representable functors to Set, this structure can often be described quite concretely in terms of "schizophrenic objects" (in the sense of Johnstone's book on "Stone Spaces"). The general theory provided here allows for a concrete computation of that group in case $ \mathcal{A} = \mathcal{B} = \mathcal{C}$ is the category of all compact and all discrete abelian groups: it is the uncountable group of algebraic automorphisms of the circle $\mathbb{R}/\mathbb{Z}$ , modulo its subgroup ${\mathbb{Z}_2}$ of continuous automorphisms.
Small solutions to inhomogeneous linear equations over number fields
Robbin
O’Leary;
Jeffrey D.
Vaaler
915-931
Abstract: We consider a system of $M$ independent, inhomogeneous linear equations in $N > M$ variables having coefficients in an algebraic number field $k$ . We give a best possible lower bound on the inhomogeneous height of a solution vector in $ {k^N}$ and determine when a solution exists in ${({\mathcal{O}_S})^N}$, where ${\mathcal{O}_S}$ is the ring of $S$-integers in $k$ . If such a system has a solution vector in $ {({\mathcal{O}_S})^N}$, we show that it has a solution $ \vec \zeta$ in ${({\mathcal{O}_S})^N}$ such that the inhomogeneous height of $\vec \zeta$ is relatively small. We give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system. Our method uses geometry of numbers over adele spaces and local to global arguments.
Algebraic cycles and the Hodge structure of a Kuga fiber variety
B. Brent
Gordon
933-947
Abstract: Let $\tilde A$ denote a smooth compactification of the $k$-fold fiber product of the universal family ${A^1} \to M$ of elliptic curves with level $ N$ structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology ${H^{\ast} }(\tilde A,\mathbb{Q})$ of $ \tilde A$ , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for $\tilde A$ ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate's conjecture concerning the algebraic cycles in the étale cohomology $H_{{\text{et}}}^{\ast} (\tilde A \otimes \bar{\mathbb{Q}},{\mathbb{Q}_l})$. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology ${H^{\ast} }({E^k},\mathbb{Q})$ of the $k$-fold product of an elliptic curve $ E$ without complex multiplication, and a verification of the generalized Hodge conjecture for ${E^k}$ .