On Tyler's M-Functional of Scatter in High Dimension

Let y¹, y²,..., yⁿ R^q be independent, identically distributed random vectors with nonsingular covariance matrix , and let S = S(y¹,..., yⁿ) be an estimator for . A quantity of particular interest is the condition number of ^-1 S. If the yⁱ are Gaussian and S is the sample covariance matrix, the condition number of ^-1 S, i.e. the ratio of its extreme eigenvalues, equals 1 + O^p((q/n)^1/2) as q and q/n 0. The present paper shows that the same result can be achieved with two estimators based on Tyler's (1987, Ann. Statist., 15, 234-251) M-functional of scatter, assuming only elliptical symmetry of (yⁱ) or less. The main tool is a linear expansion for this M-functional which holds uniformly in the dimension q. As a by-product we obtain continuous Fréchet-differentiability with respect to weak convergence.     

Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss

Let X ₁, , X _n (n > p) be a random sample from multivariate normal distribution N _p (, ), where R ^p and is a positive definite matrix, both and being unknown. We consider the problem of estimating the precision matrix ^–1. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of ^–1 is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.     

Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(, I _r) and W _p(n, ) respectively and let them be independent. The risk of the improved estimator for || or {ei329-1} based on X and S under entropy loss (=d/|| –log(d/||)–1 or d||–log(d||)–1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.Dedicated to Professor Yukihiro Kodama on his 60th birthday.     

A class of nonseparable dynamic programming problems

A solution procedure is proposed for a class of deterministic sequential decision problems whose objective functions are of the form f _n(x _n)+(g _n(x _n)) where is differentiable and either concave or convex. The procedure calls for the collaboration between dynamic programming and c-programming, and is demonstrated in our treatment of a minimum variance type problem.     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

The Symmetry of 2-Flat Homogeneous Spaces

A Riemannian manifold M is called 2-flat homogeneous if every geodesic is contained in some 2-flat , and if the group of isometries of M acts transitively on the set of pairs (p, ) with p . By a 2-flat we mean a closed, connected, flat, totally geodesic, 2-dimensional submanifold of M. It is proved in the paper that 2-flat homogeneous spaces are symmetric.     

The uniform convergence of the sum of independent random variables

LetX _i (i=1, 2, ...) be the independent random variables on the probability space (, ,P). In this paper we will show that the necessary and sufficient condition of the uniform convergence of X _i is the convergence of m _i ⁽¹⁾ and m _i ⁽⁰⁾ , wherem _i ⁽¹⁾ , m _i ⁽⁰⁾ denote the 1-quantile and 0-quantile ofX _i respectively.     

Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wⁿ P_n, where P_n is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities.     

Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen

This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO₃ ().     

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.     

Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: I. General Characteristic of the Spectrum

We study the spectrum of Hamiltonians of charged multiparticle systems in a homogeneous magnetic field with a fixed sum P of the pseudomomentum components and without it. We prove that if P is fixed, then the spectrum of Hamiltonians is independent of the value of P , while the spectrum without fixation of P coincides with the spectrum with fixation and differs from the latter only by some additional infinite degeneration (this is a principal difference between problems with a homogeneous magnetic field and problems without any field in which the absence of any fixation of the total angular momentum results in covering the spectrum of the relative motion by a continuous spectrum). We find the continuous spectrum of the Hamiltonians and characterize the spectrum of Hamiltonians of two-cluster mutually noninteracting systems obtained by decomposing the original system in the state with a fixed value of P . The last result is necessary for the study of the purely point spectrum.     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

An additive function on a ring of integers in the imaginary quadratic field Q(√d) with class-number one

LetB (a) be an additive function on a ring of integers in the quadratic number fieldQ(d) given byB (a) = _p|a *N (p) with a fixed > 0, where the asterisk means that the summation is over the non-associate prime divisorsp of an integera inQ(d), N(a) is the norm ofa. In this paper we obtain the asymptotic formula of _N(a)x *B (a) in the case where the class-number ofQ(d) is one.Project supported by the National Natural Science Foundation of China.     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

Classification de certains feuilletages associés à un cusp

We study singularities of foliations given by R. Moussu closely related to a conjecture of R. Thom. We give their analytic classification and prove their topological rigidity.

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Notations , ,F 0 - H 0, III - H ₁,H _p I - H-cusp,H _r -cusp I-3 - H _p I-3, III-2 - -cusp II-1 - _r -cusp III-1     

Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: II. The Structure of the Pure Point Spectrum

We study the pure point spectrum of the energy operator H(P ) of a many-particle charged quantum system in a homogeneous magnetic field based on the results in our previous work under fixation of the sum P of the pseudomomentum components of the system. We prove that the discrete spectrum H(P ) of a short-range system is infinite under some conditions (which, for example, hold for a system of two oppositely charged particles) even in the case of a finitely supported potential. For a long-range system of the type of a (+)-ion of an atom (including the ion), the discrete spectrum is infinite.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.