Asymptotic properties for the semiparametric regression model with randomly censored data

Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + ε, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ². It is assumed that (X,T) is independent of E. The estimators andg _n (*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that is asymptotically normal andg _n (*) is weak consistence with rateO _p(n-1/3). Project supported by China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

Between closed sets andg-closed sets

Quite recently, by using semi-open (resp.α-open, preopen,β-open) sets in a topological space, the notions ofsg*-closed (resp.αg*-closed,pg*-closedβg*-closed) sets are indroduced and investigated in [8]. These subsets place between closed sets andg-closed sets due to Levine [5]. In this paper, we introduce the notion ofmg*-closed sets and obtain the unified theory for collections of subsets between closed sets andg-closed sets.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

On the convergence of optimum stratifications for empiric distribution function in univariate case

Summary Let {X _n },n=1,2,..., be a sequence of independent random variables distributed according to a distribution functionF(x) with finite variance,F _n (x) be the empiric distribution function ofX ₁,...,X _n for eachn, andφ _(n) ^* andφ ^* be optimum stratifications corresponding toF _n (x) andF(x) respectively. It is shown in this paper thatφ _(a) ^* tends almost surely toφ ^* under a suitable criterion. Institute of Statistical Mathematics     

A splitting theorem for connected moravaK-theories

LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π_*(k(n)) ≅ ℤ _p [υ _n ] where |v _n | = 2(p ⁿ -1). We prove the following splitting theorem: Thek(n) ^*-torsion ofk(n) ^* (X) is already annihilated byv _n ^e (e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and ^r k(n) (0≤r≤e-1) where ^r k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for ^r k(n)ΛX.     

Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

Summary Let Γ=〈g ₁〉*〈g ₂〉*...*〈g _n 〉*... be a free product of cyclic groups with generators {g _i }, andC _r ^* (Γ,℘ _Λ) be the C*-algebra generated by the reduced group C*-algebraC _r ^* Γ and a set of projectionsP _gL associated with a subset Λ of {g _i }. We prove the following: (1)C _r ^* (Γ,℘ _Λ) is *-isomorphic to the reduced cross product for certain Hausdorff compact spaceX _Λ constructed from Γ and its boundary ∂Γ. (2)C _r ^* (Γ,℘ _Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the pair (Γ, Λ). (3)C _r ^* (Г,℘ _Λ) is nuclear if and only if the subgroup Γ_Λ generated by {g _i }/Λ is amenable. Partially supported by RMC grant 45/290/603 from the University of Newcastle Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S _e andS _n are independent central and noncentral Wishart matrices having Wishart distributionsW _p (n _e , Σ) andW _p (n _h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS _h S _e ⁻¹ and of certain test statistics in MANOVA under the assumption thatn=n _e +n _h becomes large with a fixed ration _e ∶n _h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).     

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g ^1/2)ωg(2 ⁿ ) mod 2 ⁿ , depending on the integersg andn, where ω _g (2 ⁿ ) denotes the order ofg modulo 2 ⁿ . Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.     

On the homology of certain spaces looped beyond their connectivity

Very little is known aboutH _*(Ωⁿ X) whenn is larger than the connectivity ofX. In this paper we calculate this whenX=Ω^∞ S ^∞ andn=1 or 2, and whenX=JU(q) or JSO(3) andn is arbitrary. Some information is also given whenX is a sphere. The authors were partially supported by the NSF.     

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution*

Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function Φ _N,β on ℝ ^M . In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m ^* (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 < γ _a < γ _c such that (1) If √α≦γ _a (m ^* (β )) ² , then the absolute minima of Φ are located within small balls around the points ± m ^* e ^μ , where e ^μ denotes the μ-th unit vector while (2) if √α≦γ _c (m ^* (β )) ² at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. Received: 5 August 1995 / In revised form: 22 May 1996     

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in ℝ ⁿ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in ℝ ⁿ . Fora ₁≥...≥a _n≥0 andq>0, denote by ℓ _w,q ⁿ then-dimensional Lorentz space with the norm ‖(x ₁,...,x _n)‖=(a ₁(x ₁ ^* ) ^q +...+a _n(x _n ^* ) ^q )^1/q , where (x ₁ ^* ,...,x _n ^* ) is the non-increasing permutation of the numbers |x ₁|,...,|x _n|. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces intoL _q [10] to prove that, forn≥3 andq≤1, the space ℓ _w,q ⁿ is isometric to a subspace ofL _q if and only if the numbersa ₁,...,a _n form an arithmetic progression. Forq>1, all the numbersa _i must be equal so that ℓ _w,q ⁿ = ℓ _q ⁿ . Consequently, the Lorentz function spaceL _w,q(0, 1) is isometric to a subspace ofL _q if and only ifeither 0<q<∞ and the weightw is a constant function (so thatL _w,q=L_q),or q≤1 andw(t) is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions. Both authors were supported in part by the NSF Workshop in Linear Analysis and Probability held at Texas A&M University in August 1993. The work was done during the first author’s visit to Texas A&M University.     

Completely positive maps and*-isomorphism ofC *-algebras II

LetA, B be unitalC ^*-algebras,D _A ¹ the set of all completely positive maps ϕ fromA toM _n (C), with Tr ϕ(I)≤1(n≥3). If Ψ is an α-invariant affine homeomorphism betweenD _A ¹ andD _B ¹ with Ψ (0)=0, thenA is^*-isomorphic toB. Obtained results can be viewed as non-commutative Kadison-Shultz theorems. This work is supported by the National Natural Science Foundation of China.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

Consistency and normality of Huber-Dutter estimators for partial linear model

For partial linear model Y = Xτβ0 g0(T) with unknown β0 ∈ Rd and an unknown smooth function g0, this paper considers the Huber-Dutter estimators of β0, scale σ for the errors and the function g0 approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β0 and σ are shown to be asymptotically normal with the rate of convergence n-1/2 and the B-spline Huber-Dutter estimator of g0 achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator.     

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.