Covering the integers by arithmetic sequences. II
Zhi-Wei
Sun
4279-4320
Abstract: Let $A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}$ ($n_{1} \leqslant \cdots \leqslant n_{k})$ be a system of arithmetic sequences where $a_{1}, \cdots ,a_{k}\in \mathbb {Z}$ and $n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}$. For $m\in \mathbb {Z}^{+}$ system $A$ will be called an (exact) $m$-cover of $\mathbb {Z}$ if every integer is covered by $A$ at least (exactly) $m$ times. In this paper we reveal further connections between the common differences in an (exact) $m$-cover of $\mathbb {Z}$ and Egyptian fractions. Here are some typical results for those $m$-covers $A$ of $\mathbb {Z}$: (a) For any $m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}$ there are at least $m$ positive integers in the form $\Sigma _{s\in I} m_{s}/n_{s}$ where $I \subseteq \{1,\cdots ,k\}$. (b) When $n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($0<l<k)$, either $l \geqslant n_{k}/n_{k-l}$ or $\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m$, and for each positive integer $\lambda <n_{k}/n_{k-l}$ the binomial coefficient $\binom l{ \lambda }$ can be written as the sum of some denominators $>1$ of the rationals $\Sigma _{s\in I}1/n_{s}- \lambda /n_{k}, I \subseteq \{1,\cdots ,k\}$ if $A$ forms an exact $m$-cover of $\mathbb {Z}$. (c) If $\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1 s\not =t}}$ is not an $m$-cover of $\mathbb {Z}$, then $\Sigma _{s\in I}1/n_{s}, I \subseteq \{1,\cdots ,k\}\setminus \{t\}$ have at least $n_{t}$ distinct fractional parts and for each $r=0,1,\cdots ,n_{t}-1$ there exist $I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}$ such that $r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}$ (mod 1). If $A$ forms an exact $m$-cover of $\mathbb {Z}$ with $m=1$ or $n_{1}< \cdots <n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($l>0$) then for every $t=1, \cdots ,k$ and $r=0,1,\cdots ,n_{t}-1$ there is an $I \subseteq \{1,\cdots ,k\}$ such that $\Sigma _{s\in I}1/n_{s} \equiv r/n_{t}$ (mod 1).
A variational problem for surfaces in Laguerre geometry
Emilio
Musso;
Lorenzo
Nicolodi
4321-4337
Abstract: We consider the variational problem defined by the functional $\int {\frac {{H^{2}-K}}{{K}}}dA$ on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.
Functorial structure of units in a tensor product
David
B.
Jaffe
4339-4353
Abstract: The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let $k$ be an algebraically closed field. Let $A$ and $B$ be reduced rings containing $k$, having connected spectra. Let $u\in A\otimes _k\,B$ be a unit. Then $u=a\otimes b$ for some units $a\in A$ and $b\in B$. Here is a deeper consequence, stated for simplicity in the affine case only. Let $k$ be a field, and let $\varphi :R\to S$ be a homomorphism of finitely generated $k$-algebras such that $\operatorname {Spec}(\varphi )$ is dominant. Assume that every irreducible component of $\operatorname {Spec}(R_{\operatorname {red}})$ or $\operatorname {Spec}(S_{\operatorname {red}})$ is geometrically integral and has a rational point. Let $B\to C$ be a faithfully flat homomorphism of reduced $k$-algebras. For $A$ a $k$-algebra, define $Q(A)$ to be $(S\otimes _k\,A)^*/(R\otimes _k\,A)^*$. Then $Q$ satisfies the following sheaf property: the sequence \begin{displaymath}0\to Q(B)\to Q(C)\to Q(C\otimes _B\,C)\end{displaymath} is exact. This and another result are used to prove (5.2) of [7].
The Morse spectrum of linear flows on vector bundles
Fritz
Colonius;
Wolfgang
Kliemann
4355-4388
Abstract: For a linear flow $\Phi$ on a vector bundle $\pi : E \rightarrow S$ a spectrum can be defined in the following way: For a chain recurrent component $\mathcal {M}$ on the projective bundle $\mathbb {P} E$ consider the exponential growth rates associated with (finite time) $(\varepsilon ,T)$-chains in $\mathcal {M}$, and define the Morse spectrum $\Sigma _{Mo}(\mathcal {M},\Phi )$ over $\mathcal {M}$ as the limits of these growth rates as $\varepsilon \rightarrow 0$ and $T \rightarrow \infty$. The Morse spectrum $\Sigma _{Mo}(\Phi )$ of $\Phi$ is then the union over all components $\mathcal {M}\subset \mathbb {P}E$. This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that $\Sigma _{Mo}(\Phi )$ contains all Lyapunov exponents of $\Phi$ for arbitrary initial values, and the $\Sigma _{Mo}(\mathcal {M},\Phi )$ are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that $\Phi$ is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of $E$ may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in $\mathbb {P} E$) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectrum. We present an example (flows induced by $C^{\infty }$ vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen's Shadowing Lemma.
The Lyapunov spectrum of families of time-varying matrices
Fritz
Colonius;
Wolfgang
Kliemann
4389-4408
Abstract: For $L^{\infty }$-families of time varying matrices centered at an unperturbed matrix, the Lyapunov spectrum contains the Floquet spectrum obtained by considering periodically varying piecewise constant matrices. On the other hand, it is contained in the Morse spectrum of an associated flow on a vector bundle. A closer analysis of the Floquet spectrum based on geometric control theory in projective space and, in particular, on control sets, is performed. Introducing a real parameter $\rho \ge 0$, which indicates the size of the $L^{\infty }$-perturbation, we study when the Floquet spectrum, the Morse spectrum, and hence the Lyapunov spectrum all coincide. This holds, if an inner pair condition is satisfied, for all up to at most countably many $\rho$-values.
Projectively bounded Fréchet measures
Ron
C.
Blei
4409-4432
Abstract: A scalar valued set function on a Cartesian product of $\sigma$-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.
Packing dimension and Cartesian products
Christopher
J.
Bishop;
Yuval
Peres
4433-4445
Abstract: We show that for any analytic set $A$ in $\mathbf {R}^d$, its packing dimension $\dim _{\mathrm {P}}(A)$ can be represented as $\; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \,$, where the supremum is over all compact sets $B$ in $\mathbf {R}^d$, and $\dim _{\mathrm {H}}$ denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if $\dim _{\mathrm {P}} (A) < d$. In contrast, we show that the dual quantity $\; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \,$, is at least the ``lower packing dimension'' of $A$, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)
The dynamical properties of Penrose tilings
E.
Arthur
Robinson Jr.
4447-4464
Abstract: The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of $% \mathbf {R}^2$ by translation. We show that this action is an almost 1:1 extension of a minimal $% \mathbf {R}^2$ action by rotations on $% \mathbf {T}^4$, i.e., it is an $% \mathbf {R}^2$ generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on $% \mathbf {T}^4$. The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.
Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields
Zhongwei
Shen
4465-4488
Abstract: We consider the Schrödinger operator with magnetic field, \begin{equation*}H=(\frac {1}{i}\nabla -{\overset {\rightharpoonup }{a}}(x))^{2}+V(x) \ \text { in } \mathbb {R}^{n}. \end{equation*} Assuming that $V\ge 0$ and $|\text {curl}\, \overset {\rightharpoonup }{a}|+V+1$ is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.
Abstract functions with continuous differences and Namioka spaces
Bolis
Basit;
Hans
Günzler
4489-4500
Abstract: Let $G$ be a semigroup and a topological space. Let $X$ be an Abelian topological group. The right differences $\triangle _{h} \varphi$ of a function $\varphi : G \to X$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\triangle _{h} \varphi$ be continuous at the identity $e$ of $G$ for all $h$ in a neighbourhood $U$ of $e$. We give conditions on $X$ or range $\varphi$ under which $\varphi$ is continuous for any topological space $G$. We also seek conditions on $G$ under which we conclude that $\varphi$ is continuous at $e$ for arbitrary $X$. This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.
Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature
Xu-Jia
Wang
4501-4524
Abstract: Let $f(x)$ be a given positive function in $R^{n+1}$. In this paper we consider the existence of convex, closed hypersurfaces $X$ so that its Gauss-Kronecker curvature at $x\in X$ is equal to $f(x)$. This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389--410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem.
The fixed-point property for simply connected plane continua
Charles
L.
Hagopian
4525-4548
Abstract: We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let $M$ be a plane continuum with the property that every simple closed curve in $M$ bounds a disk in $M$. Then every map of $M$ that sends each arc component into itself has a fixed point. Hence every deformation of $M$ has a fixed point. These results are corollaries to the following general theorem. If $M$ is a plane continuum, $\mathcal {D}$ is a decomposition of $M$, and each element of $\mathcal {D}$ is simply connected, then every map of $M$ that sends each element of $\mathcal {D}$ into itself has a fixed point.
The boundary of iterates in Euclidean growth models
Janko
Gravner
4549-4559
Abstract: This paper defines a general Euclidean growth model via a translation invariant, monotone and local transformation on Borel subsets of $\mathbf {R}^2$. The main result gives a geometric condition for the boundary curvature of the iterates to go to 0. Consequences include estimates for the speed of convergence to the asymptotic shape, and a result about survival of Euclidean deterministic forest fires.
Krull-Schmidt fails for serial modules
Alberto
Facchini
4561-4575
Abstract: We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring $R$ is a free abelian group.
Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète
Qing
Liu
4577-4610
Abstract: Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a discrete valuation field $K$. In this article we study the models of $C$ over the ring of integers $\mathcal {O}_{K}$ of $K$. To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of $C$ with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of $C$ over $\mathcal {O}_{K}$ dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over $\mathcal {O}_{K}$ of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.
Infinite products of finite simple groups
Jan
Saxl;
Saharon
Shelah;
Simon
Thomas
4611-4641
Abstract: We classify the sequences $\langle S_{n} \mid n \in \mathbb {N} \rangle$ of finite simple nonabelian groups such that $\prod$$_{n}$ $S_{n}$ has uncountable cofinality.
Curvature invariants, differential operators and local homogeneity
Friedbert
Prüfer;
Franco
Tricerri;
Lieven
Vanhecke
4643-4652
Abstract: We first prove that a Riemannian manifold $(M,g)$ with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold $(M,g)$ whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.
The stability of foliations of orientable 3-manifolds covered by a product
Sandra
L.
Shields
4653-4671
Abstract: We examine the relationship between codimension one foliations that are covered by a trivial product of hyperplanes and the branched surfaces that can be constructed from them. We present a sufficient condition on a branched surface constructed from a foliation to guarantee that all $C^1$ perturbations of the foliation are covered by a trivial product of hyperplanes. We also show that a branched surface admits a strictly positive weight system if and only if it can be constructed from a fibration over $S^1$.
Harish-Chandra's Plancherel theorem for $\frak p$-adic groups
Allan
J.
Silberger
4673-4686
Abstract: Let $G$ be a reductive $\mathfrak {p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak {p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem.
Erratum à ``Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif p-adique''
Anne-Marie
Aubert
4687-4690