On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

$\mathbb{Z}^{k}$-作用一维子系统的Lipschitz跟踪性

本文主要研究了$\mathbb{Z}^{k}$-作用一维子系统的跟踪性质. 文中运用两种等价的方式引入了$\mathbb{Z}^{k}$-作用一维子系统的伪轨以及跟踪性的概念. 对于一个闭黎曼流形上的光滑$\mathbb{Z}^{k}$-作用$T$, 我们通过诱导的非自治动力系统提出了Anosov方向的概念. 借助Bowen几何的方法, 我们证明了$T$沿着任意Anosov方向具有Lipschitz跟踪性.     

A separable non-remainder of

We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.

     

A condition for the stability of -covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be -covered. This condition can be readily verified for many examples. Further, if an -covered foliation has a compact leaf , then any transverse loop meeting lifts to a copy of the leaf space, and the ambient manifold fibers over with as fiber.

     

Centroaffine surfaces in -geodesics

For (positive) definite surfaces in there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection . We classify all surfaces in with planar -geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle.

     

A continuity property of multivariate Lagrange interpolation

Let be a sequence of interpolation schemes in of degree (i.e. for each one has unique interpolation by a polynomial of total degree and total order . Suppose that the points of tend to as and the Lagrange-Hermite interpolants, , satisfy for all monomials with . Theorem: for all functions of class in a neighborhood of . (Here denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

     

-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field are studied. It is proven that the behaviour of these maps when is not locally compact is very different from the case of real- or complex-valued functions: in general, for -compact spaces and , the existence of a (weakly) separating additive map implies that and are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of and . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

     

Arithmetic inflection formulae for linear series on hyperelliptic curves

Over the complex numbers, Plücker's formula computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in ${\mathbb{A}}^1$-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.     

over number fields and function fields

For the number field case we will give an upper bound on the number of the -integral points in
. The main tool here is the explicit upper bound of the number of solutions of -unit equations (Invent. Math. 102 (1990), 95--107). For the function field case we will give a bound on the height of the -integral points in . We will also give a bound for the number of ``generators" of those -integral points. The main tool here is the -unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427--434).

     

An Equivalent Characterization of $CMO(\mathbb{R}^n)$ with $A_p$ Weights

Let $1相似文献   

Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.     

An ascending HNN extension of a free group inside

We give an example of a subgroup of which is a strictly ascending HNN extension of a non-abelian finitely generated free group . In particular, we exhibit a free group in of rank which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold which is a surface bundle over the circle. In particular, most of comes from the fundamental group of a surface fiber. A key feature of is that there is an element of in with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group we construct is actually free.

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

Local automorphisms and derivations on certain -algebras

It is shown that continuous -local derivations on -algebras are derivations and surjective -local *-automorphisms on prime -algebras or on -algebras such that the identity element is properly infinite are *-automorphisms.

     

The -PIP and integrability of a single function

Two examples are given that answer in the negative the following question asked by E. M. Bator: If is bounded and weakly measurable and for each in there is a bounded sequence in such that a.e., does it follow that is Pettis integrable?

     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

     

Paranormal spaces under

We prove that paranormal spaces of character are -
collectionwise Hausdorff assuming the set-theoretic principle . This gives an affirmative answer to problem 197 in Problems I wish I could solve, by W. S. Watson (Open Problems in Topology (1990), 37-76).

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$-actions and the locus of Gorenstein stable surfaces

In this note, the geography of minimal surfaces of general type admitting ${\mathbb{Z}}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contains surfaces admitting a ${\mathbb{Z}}_2^2$-action for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$.