The average-shadowing property and strong ergodicity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property.     

一类超椭圆曲线的整点个数

设ｎ是大于２的工整数，Ｄ是无平方因子正整数，分别是Ｋ的理想类群和类数．对于正整数ｍ，设ｇｋ（ｍ）是Ｉｘ中阶数等于ｍ的理想类的个数．本文证明了：超椭圆曲线ｆ（ｘ，ｙ）＝Ｄｘ２－４ｙｎ＋１＝０上整数点（ｘ，ｙ）的个数不超过ｍａｘ（８，２１６４Ｐ８１ｇｋ（Ｐ）），其中ｐ是ｎ的奇素因数．     

Generalized cover ideals and the persistence property

Let I   be a square-free monomial ideal in R=k[_x1,…,_xn]$R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes Ass(I^s)$\mathrm{Ass}(I^s)$ for all integers s?1$s?1$. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G   is a tree, we explicitly determine Ass(I^s)$\mathrm{Ass}(I^s)$ for all s?1$s?1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.     

On ideals with the Rees property

A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal ${J \subset S}$ which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal ${I \subset S}$ is said to be ${\mathfrak{m}}$ -full if ${\mathfrak{m}I:y=I}$ for some ${y \in \mathfrak{m}}$ , where ${\mathfrak{m}}$ is the graded maximal ideal of ${S}$ . It was proved by one of the authors that ${\mathfrak{m}}$ -full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not ${\mathfrak{m}}$ -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.     

Uniform weak implies uniform strong persistence for non-autonomous semiflows

It is shown that, under two additional assumptions, uniformly weakly persistent semiflows are also uniformly strongly persistent even if they are non-autonomous. This result is applied to a time-heterogeneous model of S-I-R-S type for the spread of infectious childhood diseases. If some of the parameter functions are almost periodic, an almost sharp threshold result is obtained for uniform strong endemicity versus extinction.

     

A nilregular element property

An element a of a commutative ring R is nilregular if and only if x is nilpotent whenever ax is nilpotent. More generally, an ideal I of R is nilregular if and only if x is nilpotent whenever ax is nilpotent for all a ∈ I . We give a direct proof that if R is Noetherian, then every nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilregular ideals whenever R is coherent, Noetherian, and discrete. As an application, we give a constructive proof of the Eisenbud-Evans-Storch theorem that every algebraic set in n-dimensional affine space is the intersection of n hypersurfaces.Received: 6 September 2004     

On completions of non-commutative noetherian rings

It is shown that if I is an ideal of a ring R ,and I has a centralising set of generators then the I-adic completion [Rcirc] is left noetherian if either R/I is left artinian or R is left noetherian.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

关于单位正则环

本文得到了单位正则环的一个新特征,证明了:正则环R为单位正则环当且仅当存在理想I使得(1)R/I为单位正则环;(2)对任何a∈R,存在理想J满足JI=0和a=aua,其中u模J左可逆.作为应用,利用零化子理想刻画了单位正则环.     

On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property

Let X be a compact metric space and f: X → X be a continuous map. In this paper, we investigate the relationships between the asymptotic average shadowing property (Abbrev. AASP) and other notions known from topological dynamics. We prove that if f has the AASP and the minimal points of f are dense in X, then for any n ? 1, f × f × ? × f(n times) is totally strongly ergodic. As a corollary, it is shown that if f is surjective and equicontinuous, then f does not have the AASP. Moreover we prove that if f is point distal, then f does not have the AASP. For f: [0, 1] → [0, 1] being surjective continuous, it is obtained that if f has two periodic points and the AASP, then f is Li-Yorke chaotic.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

Alternating knots satisfy strong property P

Suppose a manifold is produced by finite Dehn surgery on a non-torus alternating knot for which Seifert's algorithm produces a checkerboard surface. By demonstrating that it contains an essential lamination, we prove that such a manifold has as universal cover and, consequently, is irreducible and has infinite fundamental group. Together with previous work of Roberts, who proved this result in the case of alternating knots for which Seifert's algorithm does not produce a checkerboard surface, and Moser, who classified the manifolds produced by surgery on torus knots, this paper completes the proof that alternating knots satisfy Strong Property P. Received: May 20, 1998.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Semilattices with the strong endomorphism kernel property

An endomorphism on an algebra ${\mathcal{A}}$ is said to be strong if it is compatible with every congruence on ${\mathcal{A}}$ , and ${\mathcal{A}}$ is said to have the strong endomorphism kernel property provided every congruence on ${\mathcal{A}}$ , other than the universal congruence, is the kernel of a strong endomorphism on ${\mathcal{A}}$ . In this note, we characterize those semilattices that have this property.     

On the strong fixed point property

An ordered set which has the fixed point property but not the strong fixed point property is presentedSupported by NSERC Operating Grant A7884.Supported by NSERC Operating Grant 41702.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

A note on a strong germ-Markov property

Summary In a seminal paper on Markovian germ fields, Knight [2] proposes five distinct definitions of the infinitesimal present and shows that these alternative formulations all lead to the same class of germ-Markov processes, but that they lead to different classes of strong germ-Markov processes. The same paper asserts that every germ-Markov process is strongly Markovian relative to a certain right germ field. This note will show that not every germ-Markov process (in fact, not every right continuous Markov process) is strongly germ-Markovian in such a sense. But if X=(X _t) _t0 is germ-Markovian, then the process Y defined by Y _t=(X _t, t), t0, is strongly germ-Markovian in a suitable sense. An analogous argument will settle a conjecture in [2] concerning left continuous processes and a germ-Markov property at stopping times of their natural histories not rendered right continuous.This research was supported in part by the Air Force Office of Scientific Research through its Grant No. AFOSR-80-0252.     

Faithful Inner Ideals in Jbw*-Triples

The kernel Ker(J) and the annihilator J^⊥ of a weak*-closed inner ideal J in a JBW*-triple A consist of the sets of elements a in A for which {J a J} and {J a A} are zero, respectively, and J is said to be faithful if, for every non-zero ideal I in A, I ∩ Ker (J) is non-zero. It is shown that every weak*-closed inner ideal J in A has a unique orthogonal decomposition into a faithful weak*-closed inner ideal f(J) and a weak*-closed ideal f (J)^⊥ ∩ J of A. The central structure of f ( J) is investigated and used to show that J has zero annihilator if and only if it coincides with the multiplier of f (J). The results are applied to the cases in which J is the Peirce-two or Peirce-zero space A₂(v) or A₀(v) corresponding to a tripotent v in A, and to the case in which the JBW*-triple A is a von Neumann algebra.     

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

Semigroups with the ideal retraction property

In this paper we investigate the structure of semigroups with the ideal retraction property i.e., semigroups which are not simple and have the property that each ideal is a homomorphic retract of the semigroup. We present examples to show that the ideal retraction property is neither hereditary nor productive. That this property is preserved by homomorphisms is established for some classes of semigroups, but the general question remains open. The classes of semigroups investigated in this paper are separative semigroups, ideal semigroups, semilattices, cyclic semigroups, nil semigroups, and Clifford semigroups. It is established that a semigroup with zero 0 which is expressible as a direct sum of each ideal and a dual ideal (complement with 0 adjoined) has the ideal retraction property. The converse holds for ideal semigroups, and an example is presented which demonstrates that the converse does not hold in general.