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.     

Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n$n$-cube with symmetric measure and n$n$-dimensional spherically symmetrical region. The formula forn$n$-cube contains at most n²+5n+3$n^2+5n+3$ points and for n$n$-dimensional spherically symmetrical region contains only n²+3n+3$n^2+3n+3$ points. Moreover, the numbers can be reduced to n²+3n+1$n^2+3n+1$ and n²+n+1$n^2+n+1$ if n=7$n=7$ respectively, the latter of which is minimal.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

The Helmholtz equation in a non-smooth inclusion

In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B$B$ in R²$R^2$ with boundary ∂B$\partial B$ that consists of two disjoint closed curves Γ$\Gamma$ and _Γ0${\Gamma }_0$. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ$\Gamma$ are obtained by using Riesz–Fredholm theory.     

Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

On truncated variation,upward truncated variation and downward truncated variation for diffusions

The truncated variation, TV^c${\text{TV}}^c$, is a fairly new concept introduced in ?ochowski (2008) [5]. Roughly speaking, given a càdlàg function f$f$, its truncated variation is “the total variation which does not pay attention to small changes of f$f$, below some threshold c>0$c>0$”. The very basic consequence of such approach is that contrary to the total variation, TV^c${\text{TV}}^c$ is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in ?ochowski (2011) [6], another characterization of TV^c${\text{TV}}^c$ has been found. Namely TV^c${\text{TV}}^c$ is the smallest possible total variation of a function which approximates f$f$ uniformly with accuracy c/2$c/2$. Due to these properties we envisage that TV^c${\text{TV}}^c$ might be a useful concept both in the theory and applications of stochastic processes.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].