Estimating the mean function of a Gaussian process and the Stein effect

The problem of global estimation of the mean function θ(·) of a quite arbitrary Gaussian process is considered. The loss function in estimating θ by a function a(·) is assumed to be of the form L(θ, a) = ∫ [θ(t) ? a(t)]²μ(dt), and estimators are evaluated in terms of their risk function (expected loss). The usual minimax estimator of θ is shown to be inadmissible via the Stein phenomenon; in estimating the function θ we are trying to simultaneously estimate a larger number of normal means. Estimators improving upon the usual minimax estimator are constructed, including an estimator which allows the incorporation of prior information about θ. The analysis is carried out by using a version of the Karhunen-Loéve expansion to represent the original problem as the problem of estimating a countably infinite sequence of means from independent normal distributions.     

Approximating earliest arrival flows with flow-dependent transit times

For the earliest arrival flow problem one is given a network G=(V,A) with capacities u(a) and transit times τ(a) on its arcs a∈A, together with a source and a sink vertex s,t∈V. The objective is to send flow from s to t that moves through the network over time, such that for each time θ∈[0,T) the maximum possible amount of flow up to this time reaches t. If, for each θ∈[0,T), this flow is a maximum flow for time horizon θ, then it is called earliest arrival flow. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time θ depends on the flow on this particular arc at that time θ.For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flow-dependent transit times. For that reason we define a relaxed version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each θ∈[0,T), need only α-times longer to send the maximum flow to the sink. We give both constant lower and upper bounds on α; furthermore, we present a constant factor approximation algorithm for this problem.     

Small-time sampling behavior of a Fleming-Viot process

The Fleming-Viot process with parent-independent mutation process is one particular neutral population genetic model.As time goes by,some initial species are replaced by mutated ones gradually.Once the population mutation rate is high,mutated species will elbow out all the initial species very quickly.Small-time behavior in this case seems to be the key to understand this fast transition.The small-time asymptotic results related to time scale t/θ and a(θ)t,where lim_θ→∞~(θa(θ))=0,are obtained by Dawson and Shui(1998,2001),Shui and Xiong(2002),and Xiang and Zhang(2005),respectively.Only the behavior under the scale t(θ),where lim_θ→∞~(t(θ))=0 and lim_θ→∞~(θt(θ))=∞,was left untouched.In this paper,the weak limits under various small-time scales are obtained.Of particular interest is the large deviations for the small-time transient sampling distributions,which reveal interesting phase transition.Interestingly,such a phase transition is uniquely determined by some species diversity indices.     

On differential inclusions with prescribed attainable sets

In this article, the inverse problem of the differential inclusion theory is studied. For a given ε>0 and a continuous set valued map t→W(t), t∈[t₀,θ], where W(t)⊂Rⁿ is compact and convex for every t∈[t₀,θ], it is required to define differential inclusion so that the Hausdorff distance between the attainable set of the differential inclusion at the time moment t with initial set (t₀,W(t₀)) and W(t) would be less than ε for every t∈[t₀,θ].     

Direct numerical method for an inverse problem of a parabolic partial differential equation

A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function u(x,t) and the unknown coefficient a(t) in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Decay of solutions of some nonlinear parabolic inequalities associated with time-dependent convexes

A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: lim_{t → ∞} ∝^t₀a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that lim_{t → ∞} ∝^t₀a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.     

A general approach to optimal control of a regression experiment

Let $\text{θ}\text{θ}\text{?}$ = ($\text{θ}\text{θ}\text{?}$₁, $\text{θ}\text{θ}\text{?}$₂, …, $\text{θ}\text{θ}\text{?}$_n)′ be the least-squares estimator of θ = (θ₁, θ₂, …, θ_n)′ by the realization of the process y(t) = Σ_{k = 1}ⁿθ_kf_k(t) + ξ(t) on the interval T = [a, b] with f = (f₁, f₂, …, f_n)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and D_s-optimality are used as criteria for choosing f in X. A-, D-, and D_s-optimal models can be constructed explicitly by means of r.     

A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn-Hilliard equation. It is shown in Liu, Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u(t,x) provided that the smooth nonlinear function φ(u) satisfies a local growth condition. Furthermore if φ(u) satisfies a somewhat stronger local growth condition, the optimal temporal decay estimates on u(t,x) are also obtained in Liu, Wang and Zhao (2007) [11]. Thus a natural question is how to deduce the optimal temporal decay estimates on u(t,x) only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding Cauchy problem and the main purpose of this paper is devoted to this problem. Our analysis is motivated by the technique developed recently in Ukai, Yang and Zhao (2006) [15] with a slight modification.     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

Global attractors for nonlinear wave equations with nonlinear dissipative terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x,ut)=a(x)r|ut|ut.     

A variable projection method for solving separable nonlinear least squares problems

Consider the separable nonlinear least squares problem of findinga εR ⁿ and α εR ^k which, for given data (y _i ,t _i ),i=1,2,...m, and functions ? _j (α,t),j=1,2,...,n(m>n), minimize the functional $$r(a,\alpha ) = \left\| {y - \Phi (\alpha )a} \right\|_2^2$$ where θ(α) _ij =? _j (α,t _i ). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involvingα only, and a linear least squares problem involvinga only. In this paper we propose a new method for determining the optimalα which computationally has proved more efficient than the Golub-Pereyra scheme.     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

A physical basis for Krein's prediction formula

A prediction problem of the following variety is considered. A stationary random process w(·) of known spectrum is observed over |t|?a. Using these observed values, w(b) is to be predicted for some b with |b|>a. We present a physical interpretation of a solution to this problem due to Krein, which used the theory of inverse Sturm-Liouville problems. Our physical model involves a nonuniform lossless transmission line excited at one end by white noise. The signal at the other end is the process w(t), and the prediction is found by calculating as intermediate quantities the voltage and current stored on the line at t=0. These quantities are spatially uncorrelated and provide a spatial representation at t=0 of the innovations of w(t) over |t|?a.     

Limit-point type solutions of nonlinear differential equations

We are concerned with the nonexistence of L²-solutions of a nonlinear differential equation x″=a(t)x+f(t,x). By applying technique similar to that exploited by Hallam [SIAM J. Appl. Math. 19 (1970) 430-439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L²(t₀,∞) under milder conditions on the function a(t) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes [Nonlinear Anal. 7 (1983) 851-871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations.     

Oscillation theorems for second-order nonlinear differential equations with damped term

Several oscillation criteria are given for the second-order damped nonlinear differential equation (a(t)[y′(t)]^σ′i +p(t)[y′(t)]^σ +q(t)f(y(t)) = 0, where σ > 0 is any quotient of odd integers, a ϵ C(R, (0, ∞)), p(t) and q(t) are allowed to change sign on [t_o, ∞), and f ϵ Cl (R, R) such that xf (x) > 0 for x≠0. Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.     

An asymptotically exact algorithm for one modification of planar three-index assignment problem

An m-layer three-index assignment problem is considered which is a modification of the classical planar three-index assignment problem. This problem is NP-hard for m ? 2. An approximate algorithm, solving this problem for 1 < m < n/2, is suggested. The bounds on its quality are proved in the case when the input data (the elements of an m × n × n matrix) are independent identically distributed random variables whose values lie in the interval [a _n, b _n], where b _n > a _n > 0. The time complexity of the algorithm is O(mn ² + m ^7/2). It is shown that in the case of a uniform distribution (and also a distribution of minorized type) the algorithm is asymptotically exact if m = Θ(n ^{1 ? θ} ) and b _n/a _n = o(n ^θ) for every constant θ, 0 < θ < 1.     

Asymptotic behaviour of solutions for the wave equation with an effective dissipation around the boundary

We consider the initial-boundary value problem for the wave equation with a dissipation a(t,x)u_t in an exterior domain, whose boundary meets no geometrical condition. We assume that the dissipation a(t,x)u_t is effective around the boundary and a(t,x) decays as |x|→∞. We shall prove that the total energy does not in general decay, and the solution is asymptotically free as the time goes to infinity. Further, we shall show that the local energy decays like O(t⁻¹) (t→∞).     

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η∈ (0, 1), α∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ∈ (0, λ*) under certain conditions on the nonlinear term f.