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.     

Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.     

On the sum of two closed subspaces

If U,V$U,V$ are closed subspaces of a Fréchet space, then E$E$ is the direct sum of U$U$ and V$V$ if and only if E^′$E^{\prime }$ is the algebraic direct sum of the annihilators U^°$U^{°}$ and V^°$V^{°}$. We provide a simple proof of this (possibly well-known) result.     

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.     

Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov–Bohm magnetic field

We consider two-dimensional Schrödinger operators H   with an Aharonov–Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group e^−itH$e^{-itH}$ for t→∞$t\to \infty$ in weighted L²$L^2$-spaces. In particular, we show that the magnetic field improves the decay of e^−itH$e^{-itH}$ with respect to the unitary group of non-magnetic Schrödinger operators, and that the decay rate in time is determined by the magnetic flux.     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

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].     

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$log{\delta }^{-1}$.     

Boundary value problems in spaces of distributions on smooth and polygonal domains

We study boundary value problems of the form -Δu=f$-\mathrm{\Delta }u=f$ on Ω$\Omega$ and Bu=g$\mathit{Bu}=g$ on the boundary ∂Ω$\mathrm{\partial }\Omega$, with either Dirichlet or Neumann boundary conditions, where Ω$\Omega$ is a smooth bounded domain in Rⁿ${\mathbb{R}}^n$ and the data f,g$f,g$ are distributions  . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u$u$ in terms of f and g  . For f=0$f=0$, taking advantage of the fact that u$u$ is harmonic on Ω$\Omega$, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0$f\ne 0$ and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.     

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.     

On the complexity of computing the logarithm and square root functions on a complex domain

The problems of computing single-valued, analytic branches of the logarithm and square root functions on a bounded, simply connected domain S   are studied. If the boundary ∂S$\partial S$ of S   is a polynomial-time computable Jordan curve, the complexity of these problems can be characterized by counting classes #P$\#P$, MP (or MidBitP  ), and ⊕P$\oplus P$: The logarithm problem is polynomial-time solvable if and only if FP=#P$\mathit{FP}=\#P$. For the square root problem, it has been shown to have the upper bound P^MP$P^{\mathit{MP}}$ and lower bound P^⊕P$P^{\oplus P}$. That is, if P=MP$P=\mathit{MP}$ then the square root problem is polynomial-time solvable, and if P≠⊕P$P\ne \oplus P$ then the square root problem is not polynomial-time solvable.     

Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions

We give an elementary proof for Lewis Bowen’s theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K$K$ and every free product Γ$\Gamma$ of infinite amenable groups, the factor Γ?K^Γ/K$\Gamma ?K^{\Gamma }/K$ of the Bernoulli action Γ?K^Γ$\Gamma ?K^{\Gamma }$ by the diagonal K$K$-action is isomorphic with a Bernoulli action of Γ$\Gamma$.     

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.     

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.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.