The generalized Dowling lattices
Phil
Hanlon
1-37
Abstract: In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer $n$, a finite group $G$, and a meet sublattice $K$ of the lattice of subgroups of $G$. For an appropriate choice of $ K$ the generalized Dowling lattice $ {D_n}(G,K)$ agrees with the ordinary Dowling lattice ${D_n}(G)$. For a different choice of $ K$, the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of $ G$, generalizes the thick diagonal in $ {\mathbb{C}^n}$. We compute the Möbius function and characteristic polynomial of the lattice $ {D_n}(G,K)$ along with the homology of $ {D_n}(G,K)$ in terms of the homology of $K$. We go on to compute the character of $ G$ wr ${S_n}$ acting on the homology of ${D_n}(G,K)$. This computation provides a nontrivial generalization of a result due to Stanley concerning the character of ${S_n}$ acting on the top homology of the partition lattice.
The adjoint arc in nonsmooth optimization
Philip D.
Loewen;
R. T.
Rockafellar
39-72
Abstract: We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type $ x(t) \in X(t)$ for all $ t$; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.
Strictly cyclic operator algebras
John
Froelich
73-86
Abstract: We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra $ A$ with a commutative subspace lattice is strictly cyclic iff $ \operatorname{Lat}{(A)^ \bot }$ contains a finite number of atoms and each nonzero element of $\operatorname{Lat}{(A)^ \bot }$ contains an atom. This leads to a characterization of the $ n$-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero's result that there are no triangular strictly cyclic operators. A reflexive operator algebra $ A$ with a commutative subspace lattice is strongly strictly cyclic iff $ \operatorname{Lat}(A)$ satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of $\{ 0,1] \times \{ 0,1\} \times \cdots$ which satisfy A.C.C. We also show that if $ \operatorname{Alg}(\mathcal{L})$ is strictly cyclic and $\mathcal{L} \subseteq$ atomic m.a.s.a. then $ \operatorname{Alg}(\mathcal{L})$ contains a strictly cyclic operator.
Random products of contractions in Banach spaces
J.
Dye;
M. A.
Khamsi;
S.
Reich
87-99
Abstract: We show that the random product of a finite number of $(W)$ contractions converges weakly in all smooth reflexive Banach spaces. If one of the contractions is compact, then the convergence is uniform.
Minimal submanifolds of $E\sp {2n+1}$ arising from degenerate ${\rm SO} (3)$ orbits on the Grassmannian
J. M.
Landsberg
101-117
Abstract: We give new examples of minimal submanifolds of ${{\mathbf{E}}^{2n + 1}}$ characterised by having their Gauss map's image lie in degenerate $SO(3)$ orbits of ${G_{p,2n + 1}}$, the Grassmannian of $ p$-planes in ${{\mathbf{E}}^{2n + 1}}$ (where the action on ${G_{p,2n + 1}}$ is induced from the irreducible $ SO(3)$ action on $ {{\mathbf{R}}^{2n + 1}}$). These submanifolds are all given explicitly in terms of holomorphic data and are linearly full in ${{\mathbf{E}}^{2n + 1}}$.
How porous is the graph of Brownian motion?
J. T.
Cox;
Philip S.
Griffin
119-140
Abstract: We prove that the graph of Brownian motion is almost surely porous, and determine the Hausdorff dimension of sets with a given porosity index. In particular we show that the porosity index of the graph is ${\gamma _0} \doteq 0.6948$.
Smooth dynamics on Weierstrass nowhere differentiable curves
Brian R.
Hunt;
James A.
Yorke
141-154
Abstract: We consider a family of smooth maps on an infinite cylinder which have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset which we call the "facade" of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of their invariant curves are eventually periodic.
Inner amenable locally compact groups
Anthony To Ming
Lau;
Alan L. T.
Paterson
155-169
Abstract: In this paper we study the relationship between amenability, inner amenability and property $P$ of a von Neumann algebra. We give necessary conditions on a locally compact group $G$ to have an inner invariant mean $ m$ such that $ m(V) = 0$ for some compact neighborhood $V$ of $G$ invariant under the inner automorphisms. We also give a sufficient condition on $G$ (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property $ T$, e.g., $ {\text{SL}}(n,\mathbb{Z})$, $n \geq 3$) such that each linear form on $ {L^2}(G)$ which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach $G$-modules is also obtained.
Subelliptic estimates for the $\overline \partial$-Neumann problem for $n-1$ forms
Lop-Hing
Ho
171-185
Abstract: In this note we deal with the problem of the subelliptic estimates of the $\bar \partial $-Neumann problem on nonpseudoconvex domains. In the first part we give a necessary condition for $n - 1$ forms in a class of domains. In the second part we give the exact estimate for a class of domains where the Levi form of a vector field $L$ is bounded below by a certain function.
Utility functions which ensure the adequacy of stationary strategies
Michael G.
Monticino
187-204
Abstract: Within a Dubins and Savage gambling framework, a stationary strategy is a strategy which selects a gamble at each time based solely on the gambler's present fortune. We determine conditions upon the gambler's utility function under which stationary strategies allow the gambler to maximize his return. The class of utility functions which satisfies these conditions, termed nearly leavable shift invariant functions, is large and contains many of the common gambling utility functions. Moreover, this class is closed under uniform limits. These results are obtained with the setting of an analytic gambling house.
Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games
Martino
Bardi;
Pierpaolo
Soravia
205-229
Abstract: A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games.
On completing unimodular polynomial vectors of length three
Ravi A.
Rao
231-239
Abstract: It is shown that if $ R$ is a local ring of dimension three, with $ \frac{1} {2} \in R$, then a polynomial three vector $ ({v_0}(X),{v_1}(X),{v_2}(X))$ over $R[X]$ can be completed to an invertible matrix if and only if it is unimodular. In particular, if $1/3! \in R$, then every stably free projective $R[{X_1}, \ldots ,{X_n}]$-module is free.
The Maslov class of the Lagrange surfaces and Gromov's pseudo-holomorphic curves
L. V.
Polterovich
241-248
Abstract: For an immersed Lagrange submanifold $W \subset {T^\ast }X$, one can define a nonnegative integer topologic invariant $ m(W)$ such that the image of $ {H_1}(W;{\mathbf{Z}})$ under the Maslov class is equal to $m(W) \cdot {\mathbf{Z}}$. In this paper, the value of $ m(W)$ is calculated for the case of a two-dimensional oriented manifold $ X$ with the universal cover homeomorphic to $ {{\mathbf{R}}^2}$ and an embedded Lagrange torus $W$. It is proved that if $X = {{\mathbf{T}}^2}$ and $W$ is homologic to the zero section, then $ m(W) = 0$. In all the other cases $m(W) = 2$. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in ${T^\ast }{{\mathbf{R}}^2}$. The proof is based on the Gromov's theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.
On the convergence of moment problems
J. M.
Borwein;
A. S.
Lewis
249-271
Abstract: We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak$ ^\ast$ as measures, but need not converge weakly in ${L_1}$. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in ${L_1}$. When an ${L_p}$ norm $ (1 < p < \infty)$ is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice.
New results on the Pompeiu problem
Nicola
Garofalo;
Fausto
Segàla
273-286
Abstract: Let $ {p_N}(w) = \sum\nolimits_{k = 0}^N {{a_k}{w^k}}$, $w \in \mathbb{C}$, $N \in \mathbb{N}$, be a polynomial with complex coefficients. In this paper we prove that if $D \subset {\mathbb{R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $s \in [ - \pi ,\pi ]$, then $ D$ has the Pompeiu property unless $N = 1$ and ${p_1}(w) = {a_1}w + {a_2}$ in which case $ D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.
Hyperholomorphic functions and second order partial differential equations in ${\bf R}\sp n$
R. Z.
Yeh
287-318
Abstract: Hyperholomorphic functions in ${R^n}$ with $n \geq 2$ are introduced, extending the hitherto considered hyperholomorphic functions in ${R^2}$. A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are $ (m + n - 3)!(2m + n - 2)/m!(n - 2)!$ homogeneous polynomials of degree $ m$.
Some results on the \v Sarkovski\u\i partial ordering of permutations
Irwin
Jungreis
319-344
Abstract: If $\pi$ is a cyclic permutation and $ x$ is a periodic point of a continuous function $f:{\mathbf{R}} \mapsto {\mathbf{R}}$ with ${\text{period}}(x) = {\text{order}}(\pi) = n$, then we say that $x$ has type $\pi$ if the orbit of $x$ consists of points ${x_1} < {x_2} < \cdots < {x_n}$ with $f({x_i}) = {x_{\pi (i)}}$. In analogy with Sarkovskii's Theorem, we define a partial ordering on cyclic permutations by $\theta \to \pi$ if every continuous function with a periodic point of type $\theta$ also has a point of type $\pi$. In this paper we examine this partial order form the point of view of critical points, itineraries, and kneading sequences. We show that $\theta \to \pi$ if and only if the maxima of $ \theta$ are "higher" and the minima "lower" than those of $\pi$, where "higher" and "lower" are precisely defined in terms of itineraries. We use this to obtain several results about $\to$: there are no minimal upper bounds; if $ \pi$ and $\theta$ have the same number of critical points (or if they differ by $1$ or sometimes $2$), then $ \theta \to \pi$ if and only if $ \theta \to {\pi_\ast}$ for some period double $ {\pi_\ast}$ of $ \pi$; and finally, we prove a conjecture of Baldwin that maximal permutations of size $n$ have $n - 2$ critical points, and obtain necessary and sufficient conditions for such a permutation to be maximal.
Adapted sets of measures and invariant functionals on $L\sp p(G)$
Rodney
Nillsen
345-362
Abstract: Let $G$ be a locally compact group. If $ G$ is compact, let $ L_0^p(G)$ denote the functions in ${L^p}(G)$ having zero Haar integral. Let $ {M^1}(G)$ denote the probability measures on $G$ and let ${\mathcal{P}^1}(G) = {M^1}(G) \cap {L^1}(G)$. If $ S \subseteq {M^1}(G)$, let $ \Delta ({L^p}(G),S)$ denote the subspace of ${L^p}(G)$ generated by functions of the form $f - \mu\ast f$, $ f \in {L^p}(G)$, $ \mu \in S$. If $ G$ is compact, $ \Delta ({L^p}(G),S) \subseteq L_0^p(G)$ . When $G$ is compact, conditions are given on $ S$ which ensure that for some finite subset $F$ of $S$, $\Delta ({L^p}(G),F) = L_0^p(G)$ for all $1 < p < \infty$. The finite subset $ F$ will then have the property that every $F$-invariant linear functional on ${L^p}(G)$ is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if $1 \leq p \leq \infty$, conditions are given upon $ G$, and upon subsets $ S$ of ${M^1}(G)$ whose elements satisfy certain growth conditions, which ensure that ${L^p}(G)$ has discontinuous, $ S$-invariant linear functionals. The results are applied to show that for $1 \leq p \leq \infty$, ${L^p}(\mathbb{R})$ has an infinite, independent family of discontinuous translation invariant functionals which are not $ {\mathcal{P}^1}(\mathbb{R})$-invariant.
Resonance and the second BVP
Victor L.
Shapiro
363-387
Abstract: Let $\Omega \subset {\mathbb{R}^N}$ be a bounded open connected set with the cone property, and let $1 < p < \infty$ . Also, let $ Qu$ be the $2m$th order quasilinear differential operator in generalized divergence form: $\displaystyle Qu = \sum\limits_{1 \leq \vert\alpha \vert \leq m} {{{(- 1)}^{\vert\alpha \vert}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))},$ where for $u \in {W^{m,p}}$, ${\xi _m}(u) = \{ {D^\alpha }u:\vert\alpha \vert \leq m\}$. (For $m = 1$, $Qu = - \sum\nolimits_{i = 1}^N {{A_i}(x,u,Du)}$.) Under four assumptions on $ {A_\alpha }$--Carathéodory, growth, monotonicity for $\vert\alpha \vert = m$, and ellipticity--results at resonance are established for the equation $Qu = G + f(x,u)$, where $G \in {[{W^{m,p}}(\Omega)]^\ast }$ and $f(x,u)$ satisfies a one-sided condition (plus others). For the case $m = 1$ , these results are tantamount to generalized solutions of the second BVP.
On the topology and geometric construction of oriented matroids and convex polytopes
Jürgen
Richter;
Bernd
Sturmfels
389-412
Abstract: This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety $ \mathcal{R}(M)$ of all realizations of an oriented matroid $M$ . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank $3$ oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces $\mathcal{R}(M)$ are path-connected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G. Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.
On the structure of certain locally compact topological groups
Ta Sun
Wu
413-434
Abstract: A locally compact topological group $G$ is called an $ ({\text{H}})$ group if $ G$ has a maximal compact normal subgroup with Lie factor. In this note, we study the problem when a locally compact group is an $({\text{H}})$ group.
Weak type estimates for a singular convolution operator on the Heisenberg group
Loukas
Grafakos
435-452
Abstract: On the Heisenberg group $ {\mathbb{H}^n}$ with coordinates $(z,t) \in {\mathbb{C}^n} \times \mathbb{R}$, define the distribution $K(z,t) = L(z)\delta (t)$, where $L(z)$ is a homogeneous distribution on $ {\mathbb{C}^n}$ of degree $ - 2n$ , smooth away from the origin and $ \delta (t)$ is the Dirac mass in the $t$ variable. We prove that the operator given by convolution with $K$ maps $ {H^1}({\mathbb{H}^n})$ to weak $ {L^1}({\mathbb{H}^n})$.
Effective lower bounds for the norm of the Poincar\'e $\Theta$-operator
Mark
Sheingorn
453-463
Abstract: Motivated by McMullen's proof of Kra's conjecture that the norm of the Poincaré theta operator ${\Theta _{q,\Gamma }}$ is less than $1$ for every $q$ and $\Gamma$ of finite volume, this paper provides explicit lower bounds for this norm. These bounds are sufficient to show that $\left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $\Gamma$ as $ q \to \infty$. Here the difference from $1$ is less than $ O(\frac{{{{(2\pi e)}^{q - 2}}}}{{{q^{q - 2}}}})$. For $\Gamma (N) \subseteq \Gamma \subseteq {\Gamma _0}(N)$, $\left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $q$ as $N \to \infty $. Here the difference from $1$ is $ O({N^{35 - q}})$. We prove these results by estimating $\frac{{{{\left\Vert {{\Theta _{q,\Gamma }}({f_p})} \right\Vert}_{{A_q}(\Gamma)}}}} {{{{\left\Vert {{f_p}} \right\Vert}_{{A_q}}}}}$ where the ${f_p}$ are cusp forms of weight $p \leq q - 2$. (Thus such functions may in general tend to optimize $ {\Theta _{q,\Gamma }}$.) In the case of the congruence subgroups they are taken to be products of $\Delta$ and Eisenstein series. Effective formulae are presented for all implied constants.