Quantum nonlocality for two-mode correlated states based on generalized pseudospin operators

In this Letter, we investigate the quantum nonlocality of two-mode correlated states. We find that the pseudospin formalism [Z.B. Chen, J.W. Pan, G. Hou, Y.D. Zhang, Phys. Rev. Lett. 88 (2002) 040406] generally fails to depict the nonlocality of the states when the photon number difference between the two modes is odd. The formalism is then generalized such that the nonlocality of a two-mode correlated state can be well revealed without regard to the difference. Later we consider the nonlocality of the two-mode intelligent SU(1,1) states in the generalized formalism and compare our results with the entanglement of the corresponding states.     

A separation property for magnetic Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=_ΔA+q$L={\Delta }_A+q$ on a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$, where _ΔA${\Delta }_A$ is the magnetic Laplacian on M$M$ and q≥0$q\ge 0$ is a locally square integrable function on M$M$. In the terminology of W.N. Everitt and M. Giertz, the differential expression L$L$ is said to be separated in L²(M)$L^2\left(M\right)$ if for all u∈L²(M)$u\in L^2\left(M\right)$ such that Lu∈L²(M)$Lu\in L^2\left(M\right)$, we have qu∈L²(M)$qu\in L^2\left(M\right)$. We give sufficient conditions for L$L$ to be separated in L²(M)$L^2\left(M\right)$.     

On the eigenvectors of the q‐Bernstein operators

In this article, both the eigenvectors and the eigenvalues of the q‐Bernstein operators have been studied. Explicit formulae are presented for the eigenvectors, whose limit behavior is determined both in the case 0 < q < 1 and in the case q > 1. Because the classical case, where q = 1, was investigated exhaustively by S. Cooper and S. Waldron back in 2000, the present article also discusses the related similarities and distinctions with the results in the classical case. Copyright © 2013 John Wiley & Sons, Ltd.     

The first eigenvalue of the p‐Laplace operator under powers of mean curvature flow

In this paper, we show the following main results. Let (Mⁿ,g(t)), t ∈ [0,T), be a solution of the unnormalized H^k ? flow on a closed manifold, and λ_1,p(t) be the first eigenvalue of the p‐Laplace operator. If there exists a nonnegative constant ε such that in M × [0,T) and in M × [0,T),then λ_1,p(t) is increasing and the differentiable almost everywhere along the unnormalized H^k ? flow on [0,T). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.     

Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

Error estimates for some quadrature rules with maximal trigonometric degree of exactness

In this paper, we give error estimates for quadrature rules with maximal trigonometric degree of exactness with respect to an even weight function on ( ? π,π) for integrand analytic in a certain domain of complex plane. Copyright © 2013 John Wiley & Sons, Ltd.     

On the M‐essential spectra of two‐group transport equations

In this paper, we deal with the M‐essential spectra of unbounded linear operators in Banach spaces where some generalizations of earlier work are given. Furthermore, we give an application from transport theory. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity results for infinitely many solutions for Kirchhoff‐type problems in

In this paper, we deal with existence and multiplicity of solutions of Kirchhoff‐type problems Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the preceding problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical point theory. Copyright © 2013 John Wiley & Sons, Ltd.     

Low regularity solutions,blowup, and global existence for a generalization of Camassa–Holm‐type equation

We consider a generalization of Camassa–Holm‐type equation including the Camassa–Holm equation and the Novikov equation. We mainly establish the existence of solutions in lower order Sobolev space with . Then, we present a precise blowup scenario and give a global existence result of strong solutions. Copyright © 2013 John Wiley & Sons, Ltd.