ON WELL-POSEDNESS OF THE MULTIOBJECTIVE GENERALIZED GAME

§1　IntroductionHadamard type well-posedness and Tikhonov type well-posedness are two main typesof concepts of well-posedness. At the beginning of last century,Hadamard firstintroduced the concept of well-posedness in study of optimal problem. Hadamard typewell-posedness of a problem means the continuous dependence of the solution on the dataof such problem. Later,Tikhonov introduced another concept of well-posedness.Tikhonov type well-posedness deals with the behavior ofa prescribed class …     

Extrapolation methods in numerical integration

Extrapolation methods have been used for many years for numerical integration. The most well-known of these methods is Romberg integration. A survey by Joyce on the use of extrapolation in numerical analysis appeared in 1971 in which a substantial portion is devoted to numerical integration. In this paper, we shall survey progress made in this field since 1971. The topics surveyed include partition-extrapolation methods for dealing with singular integrands, the work of Lyness and others in generating asymptotic expansions for the error functional in one and several dimensions, the work of de Doncker and others on adaptive extrapolation and the work of Sidi and others on the evaluation of highly oscillatory infinite integrals by extrapolation. Other extrapolation techniques will be mentioned briefly.     

CONDITION NUMBERS OF A MULTIPLE EIGENVALUE

This paper studies the symmetric eigenvalue problem Ax = λBx, where A,B are real symmetric matrices, and B is positive definite. Condition numbers of a multiple eigenvalue are derived from the first order perturbation expansions.     

一类滞后量为[t]的中立型泛函微分方程的渐进性和振动性

本文讨论一类滞后量为 [t]的中立型泛涵微分方程　　　　 x′(t) - c(t) x′(t- [t]+p(t) f(x(t- [t]) ) =0　 t≥ 0的解的性质 ,得到所考虑的方程存在非振动解的充分条件和非零解的变化趋势。     

The Interior Layer Problem for the Strongly Nonlinear Singularly Perturbed System

The nonlinear strongly singularly perturbed systemεyi" = fi(t, y,y'), a < t < b, i = 1, 2, …, n, yi(a,ε)-pily'i(a,ε) = Ai, yi(b,ε) pi2y'i(b,ε) = Biare considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of interior solution for the problems are studied.     

一类非线性广义Euler-Poisson-Darboux方程整体解的渐近性质

J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.     

Exact and asymptotic properties of multistate random walks

A method is presented which allows one to obtain explicit analytical expressions (both exact and asymptotic) for many of the physically interesting quantities related to a multistate random walk (MRW). The exact results include the Laplace-Fourier-transformed probability distribution (continuous time) and generating function (discrete time), and closed evolution equations for the propagators related to each internal state of the walker. Analytical expressions for the scattering dynamical structure function and the frequency-dependent diffusion coefficient are given as illustrations. Asymptotic approximations to the single-state propagators are derived, allowing a detailed analysis of the longtime behavior and the calculation of asymptotic properties by single-state random walk standard methods. As an example, analytical expressions for the drift and diffusion coefficients are given.One of the authors (M.O.C.) wants to thank the dean and research staff of the Facultad de Matemática, Astronomia y Física for their warm hospitality during his stay in Cordoba.     

Asymptotic paths for subsolutions of quasilinear elliptic equations

Letu be an entire lower semicontinuous subsolution to the quasilinear elliptic equation divA(x,u)=0 in ⁿ . It is shown that ifu is not bounded above, then there exists a path going to infinity along whichu tends to infinity. The result extends works of Talpur, Fuglede, and others. Growth aspects of subsolutions are also studied.     

Quantum shot noise: Expansions in powers of the pulse density

Quantum shot noise consists of individual pulses which contribute time-dependent (operator) potentials toward a total potentialV(t). The averaged quantity T exp _t0 ^t dtV(t) in general can no longer be calculated explicitly, in contrast to the classical case, and expansions are of interest. Noncommutative cumulant expansions are not directly applicable if the correlation functions ofV(t) have singularities, as happens in applications. It is shown here that these expansions, when applied to quantum shot noise, can be partially summed to give expansions in powers of the pulse density. Three types of such expansions are established explicitly, and for two of them the derivation is direct. For one of them the first-order approximation is closely connected to the so-called unified theory of spectral-line broadening.     

Subsample and half-sample methods

Hartigan's subsample and half-sample methods are both shown to be inefficient methods of estimating the sampling distributions. In the sample mean case the bootstrap is known to correct for skewness. But irrespective of the population, the estimates based on the subsample method, have skewness factor zero. This problem persists even if we take only samples of size less than or equal to half of the original sample. For linear statistics it is possible to correct this by considering estimates based on subsamples of size n, when the sample size is n. In the sample mean case can be taken as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeimaiaab6% cacaqG1aGaaeikaiaabgdacaqGGaGaeyOeI0IaaeiiaiaabgdacaqG% VaWaaOaaaeaacaqG1aaaleqaaOGaaeykaaaa!3E8A!\[{\text{0}}{\text{.5(1 }} - {\text{ 1/}}\sqrt {\text{5}} {\text{)}}\]. In spite of these negative results, the half-sample method is useful in estimating the variance of sample quantiles. It is shown that this method gives as good an estimate as that given by the bootstrap method. A major advantage of the half-sample method is that it is shown to be robust in estimating the mean square error of estimators of parameters of a linear regression model when the errors are heterogeneous. Bootstrap is known to give inconsistent results in this case; although, it is more efficient in the case of homogeneous errors.Research supported in part by NSA Grant MDA904-90-H-1001 and by NASA Grant NAGW-1917.