A simple technique for computing theH ∞-norm of polynomials

In this note, computation of theH -norm of polynomials is considered. It is shown that direct computation of theH -norm of polynomials, based on the definition of the norm, results in a simple and inexpensive technique for computing the norm.     

Primal-dual interior point approach for computingl 1-solutions andl ∞-solutions of overdetermined linear systems

This paper concerns solving an overdetermined linear systemA ^T x=b in the leastl ₁-norm orl -norm sense, whereA ^m×n ,m<n. We show that the primal-dual interior point approach for linear programming can be applied, in an effective manner, to linear programming versions of thel ₁ andl -problems. The resulting algorithms are simple to implement and can attain quadratic or superlinear convergence rate. At each iteration, the algorithms must solve a linear system with anm×m positive-definite coefficient matrix of the formADA ^T , whereD is a positive diagonal matrix. The preliminary numerical results indicate that the proposed algorithms offer considerable promise.This research was supported in part by Grants NSF DMS-91-02761 and DOE DE-FG05-91-ER25100.     

Explicit stationary distributions for compositions of random functions and products of random matrices

If (Y _n) _n ^=1/ is a sequence of i.i.d. random variables onE=(0,+) and iff is positive onE, this paper studies explicit examples of stationary distributions for the Markov chain (W _n) _n=0 defined byW _n=Y _nf(W _n-1). The case wheref is a Moebius function(ax+b)/(cx+d) leads to products of certain random (2,2) matrices and to interesting random continued fractions. These explicit examples are built with a naive idea by considering genral exponential families onE, especially the families of beta distributions of the first and second kind.     

Spectral Flow in Fredholm Modules, Eta Invariants and the JLO Cocycle

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer–Fredholm operators in a type I or II von Neumann algebra N. The framework is that of odd unbounded-summable Breuer–Fredholm modules for a unital Banach *-algebra, A. In the type II case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzlers idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the types I and II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the K-homology and K-theory of A.     

On the Existence of <Emphasis Type="Italic">φ</Emphasis>-Moments of the Limit of a Normalized Supercritical Galton–Watson Process

Let (Z _n ) _{n 0} be a supercritical Galton–Watson process with finite re-production mean  and normalized limit W=lim _n ^–n Z _n . Let further : [0,) [0,) be a convex differentiable function with (0)=(0)=0 and such that ( ) is convex with concave derivative for some n 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E (W) < if, and only if, , where

L ∞ Algebra Representations

It is shown that the direct sum of an L algebra and a module over it has a natural L algebra structure.     

A r (Ω)-Weighted Imbedding Inequalities for A-Harmonic Tensors

We prove the basic A _r ()-weighted imbedding inequalities for A-harmonic tensors. These results can be used to estimate the integrals for A-harmonic tensors and to study the integrability of A-harmonic tensors and the properties of the homotopy operator T: C (D, ^l )C (D, ^l–1).     

On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A _n } _n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS _n are i.i.d. exponentially distributed. (SinceA _n =0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Homotopy Theory of A∞ Ring Spectra and Applications to MU-Modules

We give a definition of a derivation of an A ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A -ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of MU-algebras where MU is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.     

L ∞ — Decay estimations of the spherical mean value on symmetric spaces

LetM _t[](x) be the spherical mean value operator applied to a function on a symmetric Riemannian space of the non-compact type.L —decay estimations forM _t [](x) as well as for its derivatives with respect to (t, x) are given, provided that belongs to a Banach space with suitable weighted supremum norm. This leads to estimates of the solutions to the wave equation in certain cases in which Huygens' principle is valid.     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

On a Lower Bound of L p Norms in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive lower bounds of the L _p norms _np for all p, 1p, in the central limit theorem for -mixing random variables with finite sixth-order moments in a strictly stationary case and finite eighth-order moments in a not necessarily stationary one.     

Robust H ∞ Filtering for a Class of Nonlinear Discrete-Time Markovian Jump Systems

This paper considers the problem of the robust H filtering for a class of nonlinear discrete-time Markovian jump systems with real time-varying norm-bounded parameter uncertainty. For each mode, the nonlinearity is assumed to satisfy the global Lipschitz conditions and appears in both the state and measured output equations. The problem that we address is the design of a nonlinear filter which ensures robust stochastic stability and a prescribed H performance level of the filtering error system for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities; an explicit expression of a desired nonlinear H filter is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field X defined on some open set M ², we characterize the prolongations X _n ^* of X to the nth jet space M ⁽ⁿ⁾, n1, such that a complete system of invariants for X _n ^* can be obtained by derivation of lower-order invariants. This leads to characterizations of C -symmetries and to new procedures for reducing the order of an ordinary differential equation.     

The Distribution Solutions of Ordinary Differential Equation with Polynomial Coefficients

Consider the Differential Equation of the form ty⁽ⁿ⁾(t) + my^(n–1)(t) + ty(t) = 0 (1) where m is any integer and n 2 for t (–, ). It is found that the values of m make the solutions of (1) to be classical, that is the solutions in the space C(–, ) of continuous functions, or the Distributions which are the solutions in the space D_R of Distributions whose supports are bounded on the left.AMS Subject Classification (1991) 46F10     

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H problem is decomposed into two lower-dimensional parameter-independent H subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

Transfer Function Optimization Procedure for the H 2/H ∞ Problem

In this paper, the H ₂/H problem is considered in a transfer-function setting, i.e., without a priori chosen bounds on the controller order. An optimization procedure is described which is based on a parametrization of all feasible descending directions stemming from a given point of the feasible transfer-function set. A search direction at each such point can be obtained on the basis of the solution of a convex finite-dimensional problem which can be converted into a LMI problem. Moving along the chosen direction in each step, the procedure in question generates a sequence of feasible points whose cost functional values converge to the optimal value of the H ₂/H problem. Moreover, this sequence of feasible points is shown to converge in the sense of a weighted H ₂ norm; and it does so to the solution of the H ₂/H problem whenever such a solution exists.