Some asymptotic expansions of moments of order statistics

Series expansions of moments of order statistics are obtained from expansions of the inverse of the distribution function. They are valid for certain types of distributions with regularly varying tails. We show that the expansions converge quickly when the sample size is moderate to large, and we obtain bounds on the rate of convergence. The special case of the Cauchy distribution is treated in more detail.     

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-input Schrödinger equation: Controllability,tracking, and application to the quantum angular momentum

We present a sufficient condition for approximate controllability of the bilinear discrete-spectrum Schrödinger equation in the multi-input case. The controllability result extends to simultaneous controllability, approximate controllability in H^s$H^s$, and tracking in modulus. The sufficient condition is more general than those present in the literature even in the single-input case and allows the spectrum of the uncontrolled operator to be very degenerate (e.g. to have multiple eigenvalues or equal gaps among different pairs of eigenvalues). We apply the general result to a rotating polar linear molecule, driven by three orthogonal external fields. A remarkable property of this model is the presence of infinitely many degeneracies and resonances in the spectrum.     

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities a(t) x ′ (t) γ′ + p 0 (t) x γ (g 0 (t)) + n i =1 p i (t) | x (g i (t)) | α i sgn x (g i (t)) = e(t), where γ is a quotient of odd positive integers, α i > 0, i = 1, 2, ··· , n, a, e, and p i ∈ C ([0, ∞ ) , R), a (t) > 0, gi : R → R are positive continuous functions on R with lim t →∞ g i (t) = ∞ , i = 0, 1, ··· , n. Our results generalize and improve the results in a recent article by Sun and Wong [29].     

Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced super-linear equation

This answers a question raised by J. S. W. Wong.

     

Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term

In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space ?². Copyright © 2007 John Wiley & Sons, Ltd.     

Oscillation of second order neutral delay differential equations of Emden-Fowler type

We present new oscillation criteria for the second order nonlinear neutral delay differential equation [y(t)-py(t-τ)]'+ q(t)y ^λ (g(t)) sgn y(g(t)) = 0, t ≧ t ₀. Our results solve an open problem posed by James S.W . Wong [24]. The relevance of our results becomes clear due to a carefully selected example. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence of probabilities for the second order monadic properties of a random mapping

We prove that the probability of each second order monadic property of a random mapping u_n converges as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 277–295, 1997     

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.     

Oscillation of second order semilinear elliptic equations with damping

We obtain some new Kamenev-type oscillation theorems for the second order semilinear elliptic differential equation with damping N ∑i,j=1Di[aij（x）Djy]＋N∑i=1bi（x）Diy＋c（x）f（y）=0 under quite general assumptions. These results are extensions of the recent results of Sun [Sun, Y. G.： New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping. J. Math. Anal. Appl., 291, 341-351 （2004）] in a natural way. In particular, we do not impose any additional conditions on the damped functions bi （x） except the continuity. Several examples are given to illustrate the main results.     

Order of convergence of second order schemes based on the minmod limiter

Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu-Tadmor scheme. It is well known that the -error of monotone finite difference methods for the linear advection equation is of order for initial data in , . For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the -error for a class of second order schemes based on the minmod limiter is of order at least in contrast to the order for any formally first order scheme.

     

Averaging technique for the oscillation of second order damped elliptic equations

By using the averaging technique, we establish some oscillation theorems for the second order damped elliptic differential equation N↑∑↓i,j=1 Di[AIY（x）Djy]＋N↑∑↓i=1 bi（x）Diy＋c（x）f（y）=0 which extend and improve some known results in the literature.     

Solving a nonlinear system of second order boundary value problems

In this paper, a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems. The analytical solution is represented in the form of series in the reproducing kernel space. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the analytical solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

Boundedness of solutions of second order nonlinear dynamic equations

This paper deals with the boundedness of the solutions of the following dynamic equations(r(t)x△(t))△+a(t)f(xσ(t))+b(t)g(xσ(t))=0and(r(t)x△(t))△+a(t)xσ(t)+b(t)f(x(t-τ(t)))=e(t)on a time scale T.By using the Bellman integral inequality,we establish some suffcient conditions for boundedness of solutions of the above equations.Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.