On Bahadur asymptotic efficiency in a semiparametric regression model

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Linear periodic control systems: Controllability with restrained controls

Consider the problem of null-controllability for a linear non-autonomous system of the form , whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht 0, and is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.     

Controllability for a class of nondiagonal hyperbolic distributed bilinear systems

This paper studies controllability of the abstract hyperbolic control systemü + Au + p(t)Bu = 0 whereA andB are (possibly) unbounded linear operators on an infinite dimensional Hilbert spaceH andp(t) is a real valued scalar control. The paper gives conditions for elements of the underlying state space to be accessible from prescribed initial data (u(0),57-1 Applications to wave equations are provided.This research was supported in part by the Air Force office of Scientific Research, Air Force Systems Command, USAF, under Contract/Grant No. AFORS-81-0172. The United States Government if authorized to reproduce and distribute reprints for government purposes not withstanding any copyright herein.     

Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side

We studyn-dimensional differential equations in distributions of the form (t)=f(x, u, t)+ b(x, u, t) (t), wheref(x, u, t) andb(x, u, t) are piecewise continuous functions andu(t) is anm-dimensional function of bounded variation with nondecreasing components. The notion of vibrosolution is introduced for equations of this type, and necessary and sufficient conditions for the existence of vibrosolutions are derived. The transition to an equivalent equation with measure is carried out, thus making it possible to explicitly calculate the jumps of the vibrosolutions at the points of discontinuity ofu(t). Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 12–21, July, 1995.     

The symmetric procrustes problem

The following symmetric Procrustes problem arises in the determination of the strain matrix of an elastic structure: find the symmetric matrixX which minimises the Frobenius (or Euclidean) norm ofAX — B, whereA andB are given rectangular matrices. We use the singular value decomposition to analyse the problem and to derive a stable method for its solution. A perturbation result is derived and used to assess the stability of methods based on solving normal equations. Some comparisons with the standard, unconstrained least squares problem are given.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

Two results on exact boundary control of parabolic equations

(1) The known nullcontrollability result for boundary control ofu _t = u_xx + q(x)u is generalized to consider a time-dependent coefficientq. (2) For boundary control ofu _t =Au (where it is known thatC _T : (initial data) (optimal nullcontrol for timeT) exists for allT>0) it is shown that logC _T =145-1 asT 0.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

Remarks on characterization of normal and stable distributions

LetX, Y, Z be independent identically distributed (i.i.d.) random variables. Suppose $$E\left| {tX + uY + vZ} \right|^p = A(\left| t \right|^q + \left| u \right|^q + \left| v \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$ for all realt, u, v, whereq=2 andp≠2m (m=1, 2,...) or 0<p<q<2. It was proved by the author this impliesX, Y, Z have the symmetricq-stable distribution. For two random variables such result is not true. One may suppose that the condition $$E\left| {tX + uY} \right|^p = A(\left| t \right|^q + \left| u \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$ and additional assumption on the behavior ofP{|X|≥x} (x→∞) implyX, Y are stable. In this paper we show it is not valid. The second result is: if the last relation holds for two different exponents andq=2, thenX andY are normal.     

Effective viscosity of an ordered suspension of small drops

We employ a method of singularity distribution to determine the effective viscosity of a suspension of small neutrally buoyant drops of one fluid in another viscous fluid. We assume that the interfacial tension is relatively large so that the drops may be considered nearly spherical and consider an instantaneous configuration in which the centers of drops coincide with the lattice points of a periodic cubic array. Under these conditions, the effective viscosity tensor is characterized by only two scalars, and, which we determine for the complete range ofc andK wherec is the volume fraction of the drops andK is the ratio of viscosities. Our numerical results, given for the simple, body-centered and face-centered cubic arrays, are in excellent agreement with those obtained for the rigid particles (K=) by Nunan and Keller except for for the face-centered cubic array where our results appear to be more accurate. The results are also in agreement with the asymptotic expressions for the dilute arrays (c1) for allK. The accuracy of the numerical results is also adequate in most cases to yield the formulas for concentrated arrays of very viscous drops (K1).

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Zusammenfassung Mit einer Singularitäten-Methode wird die effektive Viskosität einer Lösung von kleinen neutral schwimmenden Tropfen einer Flüssigkeit in einer anderen viskosen Flüssigkeit bestimmt. Wir setzen voraus daß die Oberflächenspannung verhältnismäßig groß ist, so daß man annehmen kann, daß die Tropfen fast kugelförmig sind, und wir betrachten eine Anordnung in der die Mittelpunkte der Tropfen mit den Gitterpunkten eines periodischen Kubischen Gitters zusammenfallen. Der wirksame Viskositätstensor ist in diesem Fall durch nur zwei Skalare und charakterisiert, welche wir für den gesamten Bereich vonc undK bestimmen. Hierbei istc der Volumenanteil der Tropfen, undK ist das Verhältnis der Viskositäten. Unsere numerischen Ergebnisse für die einfache kubische, raumzentriert kubische und flächenzentriert kubische Anordnung sind in ausgezeichneter Übereinstimmung mit Resultaten von Nunan und Keller für feste Teilchen (K=), ausgenommen für im flächenzentrierten System, wo unsere Resultate genauer zu sein scheinen. Die Resultate sind ebenfalls in Übereinstimmung mit asymptotischen Ausdrücken für verdünnte Suspensionen (c1) für alle Werte von K. Die Genauigkeit der numerischen Ergebnisse für konzentrierte Suspensionen von sehr viskosen Tropfen (K1) ist in den meisten Fällen ausreichend.

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On leave from Academia Sinica, Beijing, China.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.     

Products of orthogonal projections as Carleman operators

In this paper, we consider the product of two orthogonal projectionsP andQ on a separable, infinite dimensional Hilbert spaceH. For the operatorQP, there holds the dichotomy:QP is either a Carleman operator or a semi-Fredholm operator with finite defect. Both cases are characterized in terms of the dimensions of the ranges and null spaces ofP andQ and some of their intersections. This extends the case, whereP andQ are the special projections onto the subspaces of time- and band-limited functions inL ²() resp., first considered by Slepian, Pollak and Landau.     

On $${\mathcal{F}}$$ -supplemented subgroups of finite groups

Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H _G . In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized. Research of the author is supported by a NNSF grant of China (Grant #10771180).     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,) such that for some T , F ^*2(x, x+T] 2F(x, x+T]. The case T = corresponds to F being subexponential, and our analysis shows that the properties for T < are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

The martin boundary for general isotropic random walks in a tree

LetT=T _a+1 be a homogeneous tree of degreea+1. Anisotropic random walk onT is a Markov chain{X _n } such thatp(x,y)=A _d =P[X _n+1=y|X _n=x] depends only on the number of edgesd=d(x,y) betweenx andy. Assuming only thatA _d>0 for some oddd, the Martin boundary is characterized, and it is proven that nonnegativep-harmonic functions onT are the same as nearest-neighbor harmonic functions, i.e.,u(x)= _yT p(x,y)u(y) for allx T if and only ifu(x)= _d(x,y)=1 u(y)/(a)+1 for allx T. Examples are given where the Martin boundary forp is not the same as in the nearest-neighbor case. A non-Abelian renewal theory is developed such that Tauberian conditions onA _d ^* =P[d(X ₁,x)=d|X]₀=x] guarantee the equivalence of the Martin boundary. For example, the Martin boundary and Martin kernel are the same forp as for nearest-neighbor random walk if dA _d ^* < or if lim supA _d+1 ^* /A _d ^* <–1. The same techniques show the existence of renewal sequences{u _n } such thatf _k+1/f _k<- for arbitrary >1 but limu _n+1/u _n does not exist.