Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X   be a Banach lattice. We prove that every norm bounded uo-convergent net in X^?$X^{?}$ is w^?$w^{?}$-convergent iff X   has order continuous norm, and that every w^?$w^{?}$-convergent net in X^?$X^{?}$ is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ  -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w^?$w^{?}$-convergent sequence converges weakly/in norm.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

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.     

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.     

Two remarks on frequent hypercyclicity

We show that if T:X→X$T:X\to X$ is a continuous linear operator on an F$F$-space X≠{0}$X\ne \left\{0\right\}$, then the set of frequently hypercyclic vectors of T$T$ is of first category in X$X$, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→X$T:X\to X$ is a bounded linear operator on a Banach space X≠{0}$X\ne \left\{0\right\}$ and if T$T$ is frequently hypercyclic (or, more generally, syndetically transitive), then the T^∗$T^{\ast }$-orbit of every non-zero element of X^∗$X^{\ast }$ is bounded away from 0, and in particular T^∗$T^{\ast }$ is not hypercyclic.     

Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

For a Gaussian process X$X$ and smooth function f$f$, we consider a Stratonovich integral of f(X)$f\left(X\right)$, defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on X$X$ such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of f^?$f^{?}$ with respect to a Gaussian martingale independent of X$X$. The proof uses Malliavin calculus and a central limit theorem from Nourdin and Nualart (2010) [8]. This formula was known for fBm with H=1/6$H=1/6$ Nourdin et al. (2010) [9]. We extend this to a larger class of Gaussian processes.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

Global solutions to a chemotaxis system with non-diffusive memory

In this article, an existence theorem of global solutions with small initial data belonging to L¹∩L^p$L^1\cap L^p$, (n<p?∞)$(n<p?\infty )$ for a chemotaxis system is given on the whole space Rⁿ${\mathbb{R}}^n$, n?3$n?3$. In the case p=∞$p=\infty$, our global solution is integrable with respect to the space variable on some time interval, and then conserves the mass for a short time, at least. The system consists of a chemotaxis equation with a logarithmic term and an ordinary equation without diffusion term.     

A new approach to fixed point theorems on G-metric spaces

In this paper, we introduce the metric d^G$d^G$ on a G  -metric space (X,G)$(X,G)$ and use this notion to show that many contraction conditions for maps on the G  -metric space (X,G)$(X,G)$ reduce to certain contraction conditions for maps on the metric space (X,d^G)$(X,d^G)$. As applications, the proofs of many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ may be simplified, and many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ are direct consequences of preceding results for maps on the metric space (X,d^G)$(X,d^G)$.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.