Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions
Boris
Mordukhovich
1-35
Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings. The results obtained provide complete characterizations of the properties under consideration in a general setting of closed-graph multifunctions in finite dimensions. They ensure new information even in the classical cases of smooth single-valued mappings as well as multifunctions with convex graphs.
Elementary duality of modules
Ivo
Herzog
37-69
Abstract: Let $R$ be a ring. A formula $\varphi ({\mathbf{x}})$ in the language of left $ R$-modules is called a positive primitive formula (ppf) if it is of the form $ \exists {\mathbf{y}}\left({AB} \right)\left(\begin{array}{*{20}{c}}x y \end{array} \right) = 0$ where $ A$ and $B$ are matrices of appropriate size with entries in $R$. We apply Prest's notion of $D\varphi ({\mathbf{x}})$, the ppf in the language of right $R$-modules dual to $\varphi$, to show that the model theory of left $ R$-modules as developed by Ziegler [Z] is in some sense dual to the model theory of right $R$-modules. We prove that the topologies on the left and right Ziegler spectra are "isomorphic" (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism $D$ between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left $R$-module $U$ the dual pure-injective indecomposable right $R$-module $DU$. Theorem 6.6 asserts that given a complete theory $ T$ of left $R$-modules, there is a dual complete theory $DT$ of right $R$-modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable $ _RU$ which ensure that its dual $DU$ may be represented as a hom set of the form $ {\operatorname{Hom}_S}{(_R}{U_S},{E_S})$ where $S$ is some ring making $_R{U_S}$ into a bimodule and ${E_S}$ is injective.
Sieved orthogonal polynomials. VII. Generalized polynomial mappings
Jairo A.
Charris;
Mourad E. H.
Ismail
71-93
Abstract: Systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second-order difference equations are studied in detail. Applications are given to the theory of the recently discovered sieved orthogonal polynomials. The connection with polynomial mappings is examined. An example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included.
Homogeneous foliations of spheres
Duojia
Lu
95-102
Abstract: In this paper we discuss Riemannian foliations of the round sphere. We prove that there are no homogeneous Riemannian foliations of the round sphere with dimensions of the leaves bigger than three.
Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation
Atsushi
Yoshikawa
103-126
Abstract: It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data ${\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf{R}}^n}, \eta \ne 0$. Here $ {r_1}(\eta )$ is a characteristic vector, and $ {a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $n = 2$ or $3$ and that $ {a_0}(\sigma ,x)$ have the vanishing mean with respect to $\sigma$, it is shown that a genuine solution exists in a time interval independent of $ \lambda$, and that the formal solution is asymptotic to the genuine solution as $\lambda \to \infty$.
Completions and fibrations for topological monoids
Paulo
Lima-Filho
127-147
Abstract: We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion ${M^ + }$ of a monoid $M$ in that class and the topologized Grothendieck group $\tilde M$ associated to $M$. The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite $ {\text{CW}}$-complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naïve approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors $X \mapsto {\tilde{SP}}(X)$ in the category of finite $ {\text{CW}}$-complexes (Dold-Thom theorem [DT56]) is obtained.
Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes
Luis
Paris
149-178
Abstract: Let $V$ be a real vector space. An arrangement of hyperplanes in $V$ is a finite set $ \mathcal{A}$ of hyperplanes through the origin. A chamber of $\mathcal{A}$ is a connected component of $V - ({ \cup _{H \in \mathcal{A}}}H)$. The arrangement $ \mathcal{A}$ is called simplicial if ${ \cap _{H \in \mathcal{A}}}H = \{ 0\}$ and every chamber of $ \mathcal{A}$ is a simplicial cone. For an arrangement $ \mathcal{A}$ of hyperplanes in $V$, we set $\displaystyle M(\mathcal{A}) = {V_\mathbb{C}} - \left({\bigcup\limits_{H \in \mathcal{A}} {{H_\mathbb{C}}} } \right),$ where ${V_\mathbb{C}} = \mathbb{C} \otimes V$ is the complexification of $V$, and, for $ H \in \mathcal{A}$ , ${H_\mathbb{C}}$ is the complex hyperplane of ${V_\mathbb{C}}$ spanned by $H$. Let $ \mathcal{A}$ be an arrangement of hyperplanes of $V$. Salvetti constructed a simplicial complex $ \operatorname{Sal}(\mathcal{A})$ and proved that $ \operatorname{Sal}(\mathcal{A})$ has the same homotopy type as $M(\mathcal{A})$. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex $ \hat{\operatorname{Sal}}(\mathcal{A})$ and prove that there is a natural map $p:\hat {\operatorname{Sal}}(\mathcal{A}) \to \operatorname{Sal}(\mathcal{A})$ which is the universal cover of $ \operatorname{Sal}(\mathcal{A})$. At the end, we use $ \hat{\operatorname{Sal}}(\mathcal{A})$ to give a new proof of Deligne's result: "if $\mathcal{A}$ is a simplicial arrangement of hyperplanes, then $ M(\mathcal{A})$ is a $ K(\pi ,1)$ space." Namely, we prove that $ \hat{\operatorname{Sal}}(\mathcal{A})$ is contractible if $\mathcal{A}$ is a simplicial arrangement.
Multivariate discrete splines and linear Diophantine equations
Rong Qing
Jia
179-198
Abstract: In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent. In order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations. Using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent.
Set convergences. An attempt of classification
Yves
Sonntag;
Constantin
Zălinescu
199-226
Abstract: We endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics. Such structure is completely determined by a class (which is a family of closed sets) and a type (which is a semimetric). Two types are sufficient to define (and classify) almost all convergences known till now. These two types offer the possibility of defining other set convergences.
The nef value and defect of homogeneous line bundles
Dennis M.
Snow
227-241
Abstract: Formulas for the nef value of a homogeneous line bundle are derived and applied to the classification of homogeneous spaces with positive defect and to the classification of complete homogeneous real hypersurfaces of projective space.
Stability and dimension---a counterexample to a conjecture of Chogoshvili
Yaki
Sternfeld
243-251
Abstract: For every $n \geq 2$ we construct an $ n$-dimensional compact subset $X$ of some Euclidean space $E$ so that none of the canonical projections of $E$ on its two-dimensional coordinate subspaces has a stable value when restricted to $ X$. This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of $X$ and in particular its sublattice that consists of monotone decompositions when $X$ is hereditarily indecomposable.
Estimates for operator norms on weighted spaces and reverse Jensen inequalities
Stephen M.
Buckley
253-272
Abstract: We examine the dependence on the ${A_p}$ norm of $w$ of the operator norms of singular integrals, maximal functions, and other operators in ${L^p}(w)$. We also examine connections between some fairly general reverse Jensen inequalities and the ${A_p}$ and $R{H_p}$ weight conditions.
Weighted norm inequalities for Vilenkin-Fourier series
Wo-Sang
Young
273-291
Abstract: Let ${S_n}f$ be the $n$th partial sum of the Vilenkin-Fourier series of $f \in {L^1}$. For $1 < p < \infty$, we characterize all weight functions $w$ such that if $f \in {L^p}(w)$, ${S_n}f$ converges to $f$ in ${L^p}(w)$. We also determine all weight functions $w$ such that $ \{ {S_n}\}$ is uniformly of weak type $(1,1)$ with respect to $w$.
Singular integral operators on $C\sp 1$ manifolds
Jeff E.
Lewis;
Renata
Selvaggi;
Irene
Sisto
293-308
Abstract: We show that the kernel of a singular integral operator is real analytic in ${{\mathbf{R}}^n}\backslash \{ 0\}$ iff the symbol [Fourier transform] is real analytic in $ {{\mathbf{R}}^n}\backslash \{ 0\}$. The singular integral operators with continuous coefficients and real analytic kernels (symbols) form an operator algebra with the usual symbolic calculus. The symbol is invariantly defined under ${C^1}$ changes of coordinates.
Unipotent representations and reductive dual pairs over finite fields
Jeffrey
Adams;
Allen
Moy
309-321
Abstract: Consider the representation correspondence for a reductive dual pair $({G_1},{G_2})$ over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of "first occurrence", the representation correspondence takes a cuspidal unipotent representation of ${G_1}$ to one of ${G_2}$. This should serve as a fundamental case in studying the correspondence in general over both finite and local fields.
The Gorensteinness of the symbolic blow-ups for certain space monomial curves
Shiro
Goto;
Koji
Nishida;
Yasuhiro
Shimoda
323-335
Abstract: Let ${\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ denote the prime ideal in the formal power series ring $A = k[[X,Y,Z]]$ over a field $ k$ defining the space monomial curve $X = {T^{{n_1}}}$, $Y = {T^{{n_2}}}$ , and $Z = {T^{{n_3}}}$ with $ {\text{GCD}}({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebras ${R_s}({\mathbf{p}}) = { \oplus _{n \geq 0}}{{\mathbf{p}}^{(n)}}$ are Gorenstein rings for the prime ideals ${\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ with $\min \{ {n_1},{n_2},{n_3}\} = 4$ and ${\mathbf{p}} = {\mathbf{p}}(m,m + 1,m + 4)$ with $m \ne 9,13$ . The rings ${R_s}({\mathbf{p}})$ for ${\mathbf{p}} = {\mathbf{p}}(9,10,13)$ and $ {\mathbf{p}} = {\mathbf{p}}(13,14,17)$ are Noetherian but non-Cohen-Macaulay, if $\operatorname{ch}\,k = 3$ .
On a conjecture regarding nonstandard uniserial modules
Paul C.
Eklof;
Saharon
Shelah
337-351
Abstract: We consider the question of which valuation domains (of cardinality ${\aleph _1}$) have nonstandard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ${\text{ZFC}} + {\text{GCH}}$.
On automorphisms of matrix invariants
Zinovy
Reichstein
353-371
Abstract: Let ${Q_{m,n}}$ be the space of $m$-tuples of $n \times n$-matrices modulo the simultaneous conjugation action of $PG{L_n}$. Let ${Q_{m,n}}(\tau)$ be the set of points of $ {Q_{m,n}}$ of representation type $\tau$. We show that for $ m \geq n + 1$ the group $ \operatorname{Aut}({Q_{m,n}})$ of representation type preserving algebraic automorphisms of ${Q_{m,n}}$ acts transitively on each ${Q_{m,n}}(\tau)$. Moreover, the action of $ \operatorname{Aut}({Q_{m,n}})$ on the Zariski open subset ${Q_{m,n}}(1,n)$ of ${Q_{m,n}}$ is $s$-transitive for every positive integer $ s$. We also prove slightly weaker analogues of these results for all $ m \geq 3$.
Harmonic measures on covers of compact surfaces of nonpositive curvature
M.
Brin;
Y.
Kifer
373-393
Abstract: Let $M$ be the universal cover of a compact nonflat surface $N$ of nonpositive curvature. We show that on the average the Brownian motion on $M$ behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on $N$ is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.
Supercuspidal representations and Poincar\'e series over function fields
Daniel
Bump;
Shuzo
Takahashi
395-413
Abstract: In this paper, we will give a new construction of certain cusp forms on $ GL(2)$ over a rational function field. The forms which we construct are analogs of holomorphic modular forms, in that the local representations at the infinite place are in the discrete series. The novelty of our approach is that we are able to give a very explicit construction of these forms as certain 'Poincaré series.' We will also study the exponential sums which arise in the Fourier expansions of these Poincaré series.
Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials
Mohan S.
Putcha
415-428
Abstract: Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon's Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon's Hecke algebra are 'almost' the polynomials $ {R_{x,y}}$ associated with the Kazhdan-Lusztig polynomials.
Removing index $0$ fixed points for area preserving maps of two-manifolds
Edward E.
Slaminka
429-445
Abstract: Using the method of free modifications developed by M. Brown and extended to area preserving homeomorphisms, we prove the following fixed point removal theorem. Theorem. Let $h:M \to M$ be an orientation preserving, area preserving homeomorphism of an orientable two-manifold $M$ having an isolated fixed point $ p$ of index 0. Given any open neighborhood $ N$ of $ p$ such that $N \cap \operatorname{Fix}(h) = p$, there exists an area preserving homeomorphism $ \hat h$ such that (i) $\displaystyle \hat h = h\;on\;\overline {M - N} $ and (ii) $\hat h$ is fixed point free on $ N$. Two applications of this theorem are the second fixed point for the topological version of the Conley-Zehnder theorem on the two-torus, and a new proof of the second fixed point for the Poincaré-Birkhoff Fixed Point Theorem.