Equilibria and quasiequilibria for infinite collections of interacting Fleming-Viot processes
Donald A.
Dawson;
Andreas
Greven;
Jean
Vaillancourt
2277-2360
Abstract: In this paper of infinite systems of interacting measure-valued diffusions each with state space $\mathcal{P}\left( {[0,1]} \right)$, the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise as diffusion limits of population genetics models with infinitely many possible types of individuals (labelled by [0, 1]), spatially distributed over a countable collection of sites and evolving as follows. Individuals can migrate between sites and after an exponential waiting time a colony replaces its population by a new generation where the types are assigned by resampling from the empirical distribution of types at this site. It is proved that, depending on recurrence versus transience properties of the migration mechanism, the system either clusters as $t \to \infty$, that is, converges in distribution to a law concentrated on the states in which all components are equal to some ${\delta _u}, u \in [0,1]$, or the system approaches a nontrivial equilibrium state. The properties of the equilibrium states, respectively the cluster formation, are studied by letting a parameter in the migration mechanism tend to infinity and explicitly identifying the limiting dynamics in a sequence of different space-time scales. These limiting dynamics have stationary states which are quasi-equilibria of the original system, that is, change only in longer time scales. Properties of these quasi-equilibria are derived and related to the global equilibrium process for large $N$. Finally we establish that the Fleming-Viot systems are the unique dynamics which remain invariant under the associated space-time renormalization procedure.
Principally polarized ordinary abelian varieties over finite fields
Everett W.
Howe
2361-2401
Abstract: Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field $k$ to a category of ${\mathbf{Z}}$-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of ${\mathbf{Z}}$-modules. We use Deligne's equivalence to characterize the finite group schemes over $ k$ that occur as kernels of polarizations of ordinary abelian varieties in a given isogeny class over $k$. Our result shows that every isogeny class of simple odd-dimensional ordinary abelian varieties over a finite field contains a principally polarized variety. We use our result to completely characterize the Weil numbers of the isogeny classes of two-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties. We end by exhibiting the Weil numbers of several isogeny classes of absolutely simple four-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties.
Ranges of perturbed maximal monotone and $m$-accretive operators in Banach spaces
Zhengyuan
Guan;
Athanassios G.
Kartsatos
2403-2435
Abstract: A more comprehensive and unified theory is developed for the solvability of the inclusions $S \subset \overline {R(A + B)}$, int $S \subset R(A + B)$, where $A:X \supset D(A) \to {2^Y}$, $B:X \supset D(B) \to Y$ and $S \subset X$. Here, $X$ is a real Banach space and $Y = X$ or $Y = {X^*}$. Mainly, $A$ is either maximal monotone or maccretive, and $ B$ is either pseudo-monotone or compact. Cases are also considered where $ A$ has compact resolvents and $B$ is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators $A$ and $B$. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbation of maximal monotone operators, and the Leray-Schauder degree theory.
Conformal hypersurfaces with the same Gauss map
Marcos
Dajczer;
E.
Vergasta
2437-2456
Abstract: In this paper we provide a complete classification of all hypersurfaces of Euclidean space which admit conformal deformations, other than the ones obtained through conformal diffeomorphisms of the ambient space, preserving the Gauss map.
L\'evy type characterization of stable laws for free random variables
Vittorino
Pata
2457-2472
Abstract: We give a description of stable probability measures relative to free additive convolution. The definition of domain of attraction is given, and a proof is provided of the noncommutative analogue of Lévy Theorem.
Sur les structures de contact r\'eguli\`eres en dimension trois
Amine
Hadjar
2473-2480
Abstract: Let $\mathcal{M}$ be a compact and oriented $3$-manifold with boundary, endowed with a free $ {\mathbb{S}^1}$ action. We give a characterization of germs of invariant contact structures along $ \partial \mathcal{M}$ which are extendable to $ \mathcal{M}$ as regular contact structures.
Quasilinear elliptic equations with VMO coefficients
Dian K.
Palagachev
2481-2493
Abstract: Strong solvability and uniqueness in Sobolev space ${W^{2,n}}(\Omega )$ are proved for the Dirichlet problem $\displaystyle \left\{ {_{u = \varphi \quad {\text{on}}\partial \Omega .}^{{a^{i... ...{D_{ij}}u + b(x,u,Du) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\Omega }} \right.$ It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory's condition, the ${a^{ij}}$ are $V\, M\, O$ functions with respect to $x$, and structure conditions on $ b$ are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder's fixed point theorem via a priori estimate for the $ {L^{2n}}$-norm of the gradient.
On the conditional expectation and convergence properties of random sets
Nikolaos S.
Papageorgiou
2495-2515
Abstract: In this paper we study random sets, with values in a separable Banach space. First we establish several useful properties of the set-valued conditional expectation and then prove some convergence theorems for set-valued amarts and uniform amarts, using the weak, Kuratowski-Mosco and Hausdorff modes of set convergence.
On the ideal class groups of imaginary abelian fields with small conductor
Kuniaki
Horie;
Hiroko
Ogura
2517-2532
Abstract: Let $k$ be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field $ {\mathbf{Q}}$ contained in the complex field. Let $C(k)$ denote the ideal class group of $ k$, ${C^ - }(k)$ the kernel of the norm map from $ C(k)$ to the ideal class group of the maximal real subfield of $k$, and $f(k)$ the conductor of $k;f(k) \leqslant 100$. Proving a preliminary result on $2$-ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group $ {C^ - }(k)$ and, under the condition that either $[k:{\mathbf{Q}}] \leqslant 23$ or $f(k)$ is not a prime $\geqslant 71$, determines the structure of $ C(k)$.
Heegaard splittings of Seifert fibered spaces with boundary
Jennifer
Schultens
2533-2552
Abstract: We give the classification theorem for Heegaard splittings of fiberwise orientable Seifert fibered spaces with nonempty boundary. A thin position argument yields a reducibility result which, by induction, shows that all Heegaard splittings of such manifolds are vertical in the sense of Lustig-Moriah. Algebraic arguments allow a classification of the vertical Heegaard splittings.
On a semilinear elliptic Neumann problem with asymmetric nonlinearities
J.-P.
Gossez;
P.
Omari
2553-2562
Abstract: We consider the Neumann problem (1.1) below. We extend the range of applicability of the sharp nonresonance condition derived in [Go-Om] so as, in particular, allow asymmetric nonlinearities.
Characterization of summability points of N\"orlund methods
Karl-Goswin
Grosse-Erdmann;
Karin
Stadtmüller
2563-2574
Abstract: By a theorem of F. Leja any regular Nörlund method $ (N,p)$ sums a given power series $f$ at most at countably many points outside its disc of convergence. This result was recently extended to a class of non-regular Nörlund methods by K. Stadtmüller. In this paper we obtain a more detailed picture showing how possible points of summability and the value of summation depend on $ p$ and $f$.
Viewing parallel projection methods as sequential ones in convex feasibility problems
G.
Crombez
2575-2583
Abstract: We show that the parallel projection method with variable weights and one variable relaxation coefficient for obtaining a point in the intersection of a finite number of closed convex sets in a given Hilbert space may be interpreted as a semi-alternating sequential projection method in a suitably newly constructed Hilbert space. As such, convergence results for the parallel projection method may be derived from those which may be constructed in the semi-alternating sequential case.
The Toeplitz theorem and its applications to approximation theory and linear PDEs
Rong Qing
Jia
2585-2594
Abstract: We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection $ \Phi$ of compactly supported functions in $ {L_p}({\mathbb{R}^s})\quad (1 \leqslant p \leqslant \infty )$, we consider the shift-invariant space $S$ generated by $\Phi$ and the family $({S^h}:h > 0)$, where ${S^h}$ is the $h$-dilate of $S$. We prove that $ ({S^h}:h > 0)$ provides $ {L_p}$-approximation order $ r$ only if $S$ contains all the polynomials of total degree less than $r$. In particular, in the case where $ \Phi$ consists of a single function $\varphi$ with its moment $\int {\varphi \ne 0}$, we characterize the approximation order of $ ({S^h}:h > 0)$ by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.
On $C\sp *$-algebras associated to the conjugation representation of a locally compact group
Eberhard
Kaniuth;
Annette
Markfort
2595-2606
Abstract: For a locally compact group $G$, let $ {\gamma _G}$ denote the conjugation representation of $G$ in ${L^2}(G)$. In this paper we are concerned with nuclearity of ${C^*}$-algebras associated to ${\gamma _G}$ and the question of when these are of bounded representation type.
Differential identities
Bernard
Beauzamy;
Jérôme
Dégot
2607-2619
Abstract: We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
$\Pi\sp 1\sb 1$ functions are almost internal
Boško
Živaljević
2621-2632
Abstract: In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets $S$ and $T$ does there exists a bijection $f$ from $S$ onto $T$ whose graph is a projective subset of $S \times T$? In particular, when is there a $ \Pi _1^1$ bijection from $ S$ onto $T$? In this paper we prove that given an internal, bounded measure $\mu$, any $\Pi _1^1$ function is $L(\mu )$ a.e. equal to an internal function, where $ L(\mu )$ is the Loeb measure associated with $\mu$. It follows that if two $\Pi _1^1$ subsets $S$ and $T$ of a hyperfinite set $X$ are $\Pi _1^1$ bijective, then $S$ and $T$ have the same measure for every uniformly distributed counting measure $\mu$. When $S$ and $T$ are internal it turns out that any $\Pi _1^1$ bijection between them must already be Borel. We also prove that if a $ \Pi _1^1$ graph in the product of two hyperfinite sets $X$ and $Y$ is universal for all internal subsets of $ Y$, then $\vert X\vert \geqslant {2^{\vert Y\vert}}$, which is a partial answer to Henson and Ross's Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper $K$-analytic subsets of the product of two completely regular Hausdorff topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of $ \Pi _1^1$ subsets of the products $X \times Y$ of two internal sets $ X$ and $Y$, all of whose $Y$-sections are $\Sigma _1^0(\kappa )$ sets.
On a singular quasilinear anisotropic elliptic boundary value problem
Y. S.
Choi;
A. C.
Lazer;
P. J.
McKenna
2633-2641
Abstract: We consider the problem $\displaystyle {u^a}{u_{xx}} + {u^b}{u_{yy}} + p({\mathbf{x}}) = 0$ with $a \geqslant 0$, $b \geqslant 0$, on a smooth convex bounded region in ${{\mathbf{R}}^2}$ with Dirichlet boundary conditions. We show that if the positive function $ p$ is uniformly bounded away from zero, then the problem has a classical solution.
Affine transformations and analytic capacities
Thomas
Dowling;
Anthony G.
O’Farrell
2643-2655
Abstract: Analytic capacities are set functions defined on the plane which may be used in the study of removable singularities, boundary smoothness and approximation of analytic functions belonging to some function space. The symmetric concrete Banach spaces form a class of function spaces that includes most spaces usually studied. The Beurling transform is a certain singular integral operator that has proved useful in analytic function theory. It is shown that the analytic capacity associated to each Beurling-invariant symmetric concrete Banach space behaves reasonably under affine transformation of the plane. It is not known how general analytic capacities behave under affine maps.
Involutory Hopf algebras
D. S.
Passman;
Declan
Quinn
2657-2668
Abstract: In 1975, Kaplansky conjectured that a finite-dimensional semisimple Hopf algebra is necessarily involutory. Twelve years later, Larson and Radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics. The goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites, no "harpoons", and gratifyingly few "hits".
Variational formulas on Lipschitz domains
Alan R.
Elcrat;
Kenneth G.
Miller
2669-2678
Abstract: A rigorous treatment is given of variational formulas for solutions of certain Dirichlet problems for the Laplace operator on Lipschitz domains under interior variations. In particular we extend well-known variational formulas for the torsional rigidity and for capacity from the class of $ {C^1}$ domains to the class of Lipschitz domains. A motivation for this work comes from the use of variational methods in the study of Prandtl-Batchelor flows in fluid mechanics.
Compact composition operators on the Bloch space
Kevin
Madigan;
Alec
Matheson
2679-2687
Abstract: Necessary and sufficient conditions are given for a composition operator $ {C_\phi }f = f{\text{o}}\phi$ to be compact on the Bloch space $\mathcal{B}$ and on the little Bloch space ${\mathcal{B}_0}$. Weakly compact composition operators on $ {\mathcal{B}_0}$ are shown to be compact. If $\phi \in {\mathcal{B}_0}$ is a conformal mapping of the unit disk $ \mathbb{D}$ into itself whose image $ \phi (\mathbb{D})$ approaches the unit circle $ \mathbb{T}$ only in a finite number of nontangential cusps, then ${C_\phi }$ is compact on ${\mathcal{B}_0}$. On the other hand if there is a point of $ \mathbb{T} \cap \phi (\mathbb{D})$ at which $ \phi (\mathbb{D})$ does not have a cusp, then ${C_\phi }$ is not compact.