Beurling-Lax representations using classical Lie groups with many applications II: GL(n,©) and Wiener-Hopf factorization

The classical Beurling-Lax-invariant subspace theorem characterizes the full range simply invariant subspacesM of L _n ² as being of the formM=H _n ² where L _n×n is a phase function. Here L _n ² is the Hilbert space of measurable ⁿ-valued functions on the unit circle {e^it|0t2} which are square-integrable in norm, H _n ² is the subspace of functions in L _n ² with analytic continuation to the interior of the disk {zz|<1}, L _n×n ² is the space of measurable essentially bounded n×n matrix functions on the unit circle, and a phase function is one whose values (e^it) are unitary for a.e. t (i.e., (e^it) is in the Lie group U(n) a.e.). Halmos extended this to L ² . A subspace ML _n ² is said to beinvariant if e^it MM,simply invariant if in addition e^ikt M=(0), andfull range if 0} $$ " align="middle" border="0"> e^–iNt M is dense in L _n ² . In the Beurling-Lax representationM=H _n ² ,M uniquely determines up to a unitary constant factor on the right if one insists that (e^it)U(n). If one demands only that (e^it) GL(n,) (the group of invertible n×n complex matrices), however, there is considerably more freedom; in fact H _n ² =₁H _n ² where ₁ F and FL _n×n is outer with inverse F^–1L _n×n . More generally, we have H _n ² =[₁H _n ]^– whenever ₁=F and F is outer with F and F^–1 in L _n×n ² . (An FL _n×n ² will be said to beouter if FH _n is a dense subset of H _n ² .) In particular one can use this freedom to obtain representationsM=[H _n ]^– where the representor has values (e^it) in other matrix Lie groups. This program was carried out in accompanying work of the authors [B-H1-4] for the classical simple Lie groups U(m,n), O(p,q), O^*(2n), Sp(n,C), Sp(n,R), Sp(p,q), O(n,C), GL(n,R), U^*(2n), GL(n,R) and SL(n,C) and many applications were given. In this paper we give a natural theorem for GL(n,), by introducing the extra structure of preassigning the spaceM ^x=[H _n ]^– as well asM=[H _n ]^–. The theorems in [B-H1-4] can be derived by specializing our main result here for GL(n,) to the various subgroups which we listed.Both authors are partially supported by the National Science Foundation.     

Description of Hyperinvariant Subspaces of a Contraction in Terms of Its Characteristic Function

If T is a completely nonunitary contraction on a Hilbert space and L is its invariant subspace corresponding to a regular factorization of its characteristic function = , then L is hyperinvariant if and only if the following two conditions are fulfilled: (1) supp _* supp is of Lebesgue measure zero; (2) for every pair A H (E E) and A _* H (E _* E _*) intertwining by , i.e., such that A =A _*, there exists a function A _F H (F F) intertwining with A by and with A _* by , i.e., such that A = A _F and A _F = A _*. Bibliography: 4 titles.     

Information Inequalities in a Family of Uniform Distributions

For a family of uniform distributions, it is shown that for any small < 0 the average mean squared error (MSE) of any estimator in the interval of values of length and centered at ₀ can not be smaller than that of the midrange up to the order o(n ^–2) as the size n of sample tends to infinity. The asymptotic lower bound for the average MSE is also shown to be sharp.     

Estimating Equations with Nuisance Parameters: Theory and Applications

In a variety of statistical problems the estimate _n of a parameter is defined as the root of a generalized estimating equation G_n(_nn)=0 where _n is an estimate of a nuisance parameter . We give sufficient conditions for the asymptotic normality of #x0398;_n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;_n will not be influenced by that of _n) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.     

Efficient robust estimates in parametric models

Summary Let {P ⁿ :}, an open subset ofR ^k , be a regular parametric model for a sample ofn independent, identically distributed observations. This paper describes estimates {T _n ;n1} of which are asymptotically efficient under the parametric model and are robust under small deviations from that model. In essence, the estimates are adaptively modified, one-step maximum likelihood estimates, which adjust themselves according to how well the parametric model appears to fit the data. When the fit seems poor,T _n discounts observations that would have large influence on the value of the usual one-step MLE. The estimates {T _n } are shown to be asymptotically minimax, in the Hájek-LeCam sense, for a Hellinger ball contamination model. An alternative construction of robust asymptotically minimax estimates, as modified MLE's, is described for canonical exponential families.This research was supported in part by National Science Foundation Grant MCS 75-10376     

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

Locally most powerful rank tests for testing randomness and symmetry

Let X _i, 1 i N, be N independent random variables (i.r.v.) with distribution functions (d.f.) F _i(x,), 1 i N, respectively, where is a real parameter. Assume furthermore that F _i(·,0) = F(·) for 1 i N. Let R = (R ₁,R _N) and R ⁺,...,R _N ⁺ be the rank vectors of X = (X ₁,X _N) and |X|=(|X ₁|,...,|X _N|), respectively, and let V = (V ₁,V _N) be the sign vector of X. The locally most powerful rank tests (LMPRT) S = S(R) and the locally most powerful signed rank tests (LMPSRT) S = S(R ⁺, V) will be found for testing = 0 against > 0 or < 0 with F being arbitrary and with F symmetric, respectively.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A2 ^S : W(A,P) _S (d(x,A))P(dx)min _AA , where is a nondecreasing function. A special case where A is a parametric class A={A():T} is considered in detail. Our main interest is to obtain convergence results for sequences {A ^* _n }, where A ^* _n is an optimal set for a measure P _n satisfying P _n P, as n.     

Factorization of transfer functions. I. (+).regular factorization

In the first part of the paper the concept of (+).regular factorization is generalized to the case of n factors.THEOREM 1. The factorization =_nn–1...₁, is (+).regular where (_k) is the Sz.-Nagy-Foias space. A criterion for the preservation of minimality under the synthesis of conservative scattering systems is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 312–317, March, 1990.     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

Isotone Maps as Maps of Congruences. II. Concrete Maps

Let L be a lattice and let L ₁, L ₂ be sublattices of L. Let be a congruence relation of L ₁. We extend to L by taking the smallest congruence......     

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Probabilistic analysis of an algorithm for solving thek-dimensional all-nearest-neighbors problem by projection

A well-known simple heuristic algorithm for solving the all-nearest-neighbors problem in thek-dimensional Euclidean spaceE ^k ,k>1, projects the given point setS onto thex-axis. For each pointq S a nearest neighbor inS under anyL _p -metric (1 p ) is found by sweeping fromq into two opposite directions along thex-axis. If _q denotes the distance betweenq and its nearest neighbor inS the sweep process stops after all points in a vertical 2 _q -slice centered aroundq have been examined. We show that this algorithm solves the all-nearest-neighbors problem forn independent and uniformly distributed points in the unit cube [0,1] ^k in (n ^2–1/k ) expected time, while its worst-case performance is (n ²).     

Maximal packing density of n-dimensional Euclidean space with equal balls

An estimate _n 2^{-n(0.5237+o(1))} is obtained for the maximal packing density of n-dimensional Euclidean space with equal balls for n . This result is based on an improvement in a corresponding upper estimate for the maximal packing density of the unit (n-1)-dimensional sphere with spherical caps of fixed angular radius.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 301–311, August, 1975.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.