Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

On topological entropy of commuting tree maps

Let T$T$ be a tree with s$s$ ends and f,g$f,g$ be continuous maps from T$T$ to T$T$ with f°g=g°f$f°g=g°f$. In this note we show that if there exists a positive integer m≥2$m\ge 2$ such that gcd(m,l)=1$gcd(m,l)=1$ for any 2≤l≤s$2\le l\le s$ and f,g$f,g$ share a periodic point which is a km$km$-periodic point of f$f$ for some positive integer k$k$, then the topological entropy of f°g$f°g$ is positive.     

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α$\alpha$ is in (0,2)$(0,2)$, equal to 2, and in (2,∞)$(2,\infty )$, respectively. The partial sum weakly converges to a functional of α$\alpha$-stable process when α<2$\alpha <2$ and converges to a functional of Brownian motion when α≥2$\alpha \ge 2$. When the process is of short-memory and α<4$\alpha <4$, the autocovariances converge to functionals of α/2$\alpha /2$-stable processes; and if α≥4$\alpha \ge 4$, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α$\alpha$ and β$\beta$ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2$\alpha /2$-stable processes; (ii) Rosenblatt processes (indexed by β$\beta$, 1/2<β<3/4$1/2<\beta <3/4$); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α$\alpha$ and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1]$[0,1]$ with either (i) the _J1$J_1$ or the _M1$M_1$ topology (Skorokhod, 1956); or (ii) the weaker form S$S$ topology (Jakubowski, 1997). Some statistical applications are also discussed.     

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f$f$ of a strictly pseudoconvex hypersurface M⊂Cⁿ⁺¹$M\subset {\mathbb{C}}^{n+1}$ into a sphere S$\mathbb{S}$ in a higher dimensional complex space C^N+1${\mathbb{C}}^{N+1}$. The degeneracy of the mapping f$f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f$f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d$d$ of the second fundamental form and all of its covariant derivatives is <n$<n$ (here, n$n$ is the CR dimension of M$M$), then f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank d$d$ exceeds n$n$, it is no longer true, in general, that f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n$n$, then partial rigidity may still persist, but there is a “defect” k$k$ that arises from the ranks exceeding n$n$ such that f(M)$f\left(M\right)$ is only contained in a complex plane of dimension n+d+k+1$n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.     

Modules satisfying the weak Nakayama property

Let R$R$ be a commutative ring with identity. We will say that an R$R$-module M$M$ satisfies the weak Nakayama property, if IM=M$IM=M$, where I$I$ is an ideal of R$R$, implies that for any x∈M$x\in M$ there exists a∈I$a\in I$ such that (a−1)x=0$(a-1)x=0$. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R$R$ is a local ring, then R$R$ is a Max ring if and only if J(R)$J\left(R\right)$, the Jacobson radical of R$R$, is T$T$-nilpotent if and only if every R$R$-module satisfies the weak Nakayama property.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

A note on Schweizer–Smital chaos

In this paper, we consider a continuous map f:X→X$f:X\to X$, where X$X$ is a compact metric space, and prove that for any positive integer N$N$, f$f$ is Schweizer–Smital chaotic if and only if f^N$f^N$ is too.     

Finiteness conditions on the Yoneda algebra of a monomial algebra

Let A$A$ be a connected graded noncommutative monomial algebra. We associate to A$A$ a finite graph Γ(A)$\Gamma \left(A\right)$ called the CPS graph of A$A$. Finiteness properties of the Yoneda algebra Ext_A(k,k)${\text{Ext}}_A(k,k)$ including Noetherianity, finite GK dimension, and finite generation are characterized in terms of Γ(A)$\Gamma \left(A\right)$. We show that these properties, notably finite generation, can be checked by means of a terminating algorithm.     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.