Prediction for Two Processes and the Nehari Problem

We exploit an analogy between the trigonometric moment problem and prediction theory for a stationary stochastic process. Extending this theory, we show how to use correlations between two processes to predict one from the other. In turn, this gives rise to a simple and unified treatment of the Caratheodory and Nehari moment problems.     

Prediction for two processes and the nehari problem

We exploit an analogy between the trigonometric moment problem and prediction theory for a stationary stochastic process. Extending this theory, we show how to use correlations between two processes to predict one from the other. In turn, this gives rise to a simple and unified treatment of the Caratheodory and Nehari moment problems.     

Krein's strings,the symmetric moment problem,and extending a real positive definite function

The symmetric moment problem is to find a possibly unique, positive symmetric measure that will produce a given sequence of moments {M_n}. Let us assume that the (Hankel) condition for existence of a solution is satisfied, and let σ_n be the unique measure, supported on n points, whose first 2n moments agree with M₀,…,M_2n−1. It is known that σ_2n ⇒ σ₀ (weak convergence) and σ_2n+1 ⇒ σ_∞, where σ₀ and σ_∞ are solutions to the full moment problem. Moreover, σ₀ = σ_∞ if and only if the problem has a unique solution. In this paper we present an analogue of this theorem for Krein's problem of extending to ℝ a real, even positive definite function originally defined on [−T,T] where T < ∞. Our proof relies on the machinery of Krein's strings. As we show, these strings help explain the connection between the moment and the extension problems. © 1999 John Wiley & Sons, Inc.     

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height M > 0 [4] in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non–radial) minimizer in accordance with the results of [9]. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with n sides centered in the disc, and numerical experiments indicate that the natural number n > 2 is a non–decreasing function of M. The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height M.     

Sampling Integrodifferential Transforms Arising from Second Order Differential Operators

We give sampling theorems associated with boundary value problems whose differential equations are of the form M(y) = λS(y), where M and S are differential expressions of the second and first order respectively and the eigenvalue parameter may appear in the boundary conditions. The class of the sampled functions is not a class of integral transforms as is the case in the classical sampling theory, but it is a class of integrodifferential transforms. We use solutions of the problem as well as Green's function to derive two sampling theorems.     

Aerodynamic Sound Emission as a Singular Perturbation Problem

Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ? 1, Λ ? 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment T_ij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ? × ψ{ ω } + ?φ The problem is solved for M ? 1, Λ ? 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if T_ij is approximated by ρ₀υ_iυ_j, with v = ? × ψ. The situation M ? 1, Λ ? 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.     

Moment convergence of <Emphasis Type="Italic">Z</Emphasis>-estimators

The problem to establish the asymptotic distribution of statistical estimators as well as the moment convergence of such estimators has been recognized as an important issue in advanced theories of statistics. This problem has been deeply studied for M-estimators for a wide range of models by many authors. The purpose of this paper is to present an alternative and apparently simple theory to derive the moment convergence of Z-estimators. In the proposed approach the cases of parameters with different rate of convergence can be treated easily and smoothly and any large deviation type inequalities necessary for the same result for M-estimators do not appear in this approach. Applications to the model of i.i.d. observation, Cox’s regression model as well as some diffusion process are discussed.     

Concentration on minimal submanifolds for a Yamabe-type problem

We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.     

On boundary controllability of one-dimensional vibrating systems by W,p-controls for p ∈ [2, ∞]

This paper is concerned with boundary control of one-dimensional vibrating media whose motion is governed by a wave equation with a 2n-order spatial self-adjoint and positive-definite linear differential operator with respect to 2n boundary conditions. Control is applied to one of the boundary conditions and the control function is allowed to vary in the Sobolev space W, ^p for p∈[2, ∞] With the aid of Banach space theory of trigonometric moment problems, necessary and sufficient conditions for null-controllability are derived and applied to vibrating strings and Euler beams. For vibrating strings also, null-controllability by L^p-controls on the boundary is shown by a direct method which makes use of d'Alembert's solution formula for the wave equation.     

Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation ,

we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies

||p||_L2(K)An^1/2 and ||p′||_L2(K)Bn^3/2.

Then

M₄(p)−M₂(p)M₂(p)

with

We also prove that

and

M₂(p)−M₁(p)10⁻³¹M₂(p)

for every , where denotes the collection of all trigonometric polynomials of the form

     

Stieltjes Moment Problems and the Friedrichs Extension of a Positive Definite Operator

For an indeterminate Stieltjes moment sequence the multiplication operator Mp(x) = xp(x) is positive definite and has self-adjoint extensions. Exactly one of these extensions has the same lower bound as M, the so-called Friedrichs extension. The spectral measure of this extension gives a certain solution to the moment problem and we identify the corresponding parameter value in the Nevanlinna parametrization of all solutions to the moment problem. In the case where σ is indeterminate in the sense of Stieltjes, relations between the (Nevanlinna matrices of) entire functions associated with the measures t^kdσ(t) are derived. The growth of these entire functions is also investigated.     

Index policies for the maintenance of a collection of machines by a set of repairmen

A collection of M machines which deteriorate under usage is maintained by a set of R repairmen who may have other tasks to perform. Maintenance interventions will improve a machine's condition and may preempt costly breakdowns. The problem of scheduling such interventions to minimise the total expected discounted cost incurred in operating the machines over an infinite horizon is formulated as a Markov Decision Process which has the form of a restless bandit problem. We outline an approach to the development of maintenance policies based on simple machine indices in the form of a fair charge for a maintenance intervention in the machine's current state. Closed form indices are derived for two particular models. Numerical investigations demonstrate the strong performance of the derived index heuristics.     

Filtering,Smoothing and M-ary Detection with Discrete Time Poisson Observations

Abstract

In this article, we solve a class of estimation problems, namely, filtering smoothing and detection for a discrete time dynamical system with integer-valued observations. The observation processes we consider are Poisson random variables observed at discrete times. Here, the distribution parameter for each Poisson observation is determined by the state of a Markov chain. By appealing to a duality between forward (in time) filter and its corresponding backward processes, we compute dynamics satisfied by the unnormalized form of the smoother probability. These dynamics can be applied to construct algorithms typically referred to as fixed point smoothers, fixed lag smoothers, and fixed interval smoothers. M-ary detection filters are computed for two scenarios: one for the standard model parameter detection problem and the other for a jump Markov system.     

Random invariant measures for Markov chains,and independent particle systems

Summary Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.Research supported in part by N.S.F. Grant No. MCS 77-02121     

Carr's randomization for American options in regime-switching models

In the paper, we solve the pricing problem for American put-like options in Markov-modulated Lévy models. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization for regime-switching models. An efficient iteration pricing procedure is developed. The computational time is of order m², where m is the number of states, and of order m, if the parallel computations are allowed. The payoffs, riskless rates and class of Lévy processes may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Markov renewal queues of M/M(a,d)/l/(b,n) system-part i

This paper discusses a general bulk service queue which falls into the Markov renewal class. Applying an analysis similar to the one by Hunter (1983) for M/M1/N type of feedback queues, certain properties of discrete and continuous time queue length processe are studied here. The results and formulas are then applied to a numerical illustration.     

Numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm

This paper deals with an inverse problem of identifying a nonlinear source term g=g(u) in the heat equation u_t-u_xx=a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M>0, with which an optimal perturbation algorithm is applied to determine the source function g(u) on u∈[0,M]. Numerical inversions are carried out for g(u) with functional forms of polynomial, trigonometric and index functions. The inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here. By the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources. The inversion algorithm seems to be very sharp if the solution’s maximum M of the forward problem is relatively small; otherwise, the deviations in the source solutions become large especially near the endpoint of u=M.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

Torsion in the full orbifold <Emphasis Type="Italic">K</Emphasis>-theory of abelian symplectic quotients

Let (M, ω, Φ) be a Hamiltonian T-space and let H í T{H\subseteq T} be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S ¹-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.