QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

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Subgroups of the groupZ n and a generalization of the Gołąb-Schinzel functional equation

Letn be a positive integer and letX be a linear space over a commutative fieldK. In the set = (K\{0}) × X we define a binary operation ·: × by

Strong topologies on vector-valued function spaces

Let be a real Banach space and let E be an ideal of L ⁰ over a -finite measure space (, , ). Let (X) be the space of all strongly -measurable functions f: X such that the scalar function , defined by , belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.     

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

M-estimators of structural parameters in pseudolinear models

Real valued M-estimators in a statistical model 1 with observations are replaced by -valued M-estimators in a new model with observations where are regressors, is a structural parameter and a structural function of the new model. Sufficient conditions for the consistency of are derived, motivated by the sufficiency conditions for the simpler parent estimator The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If F has a natural exponential density e^{x–b( x )} then our pseudolinear model with u = (g o )^–1 reduces to the well known generalized linear model, provided () = db()/d and g is the so-called link function of the generalized linear model. General results are illustrated for special pairs and leading to some classical M-estimators of mathematical statistics, as well as to a new class of generalized -quantile estimators.     

The convergence rate for the strong law of large numbers: General lattice distributions

LetX ₁,X ₂, ... be a sequence of independent random variables with common lattice distribution functionF having zero mean, and let (S _n ) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any and >0

On the divergence of linear summation methods of Fourier series

[0, 1],fL(0,2),

Wilson's functional equations on groups

We study properties of solutionsf, g, h C(G) of the functional equation

The normal density of prime ideals in small regions

For a Gaussian prime (i) define ()=min|–| where runs through Gaussian primes satisfying ||>||. We prove that, subject to the Riemann Hypothesis for appropriateL-functions

On the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.     

Quasirationale singularitäten

It Is shown that Elkik's results on deformations of rational singularities hold without Cohen-Macaulay assumptions: let f:XS be flat with normal fibres X_s and desingularisations _s:Y_sX_s; then the sets are open for all d.     

Spherical elastic-plastic waves

Zusammenfassung Es wird die Fortpflanzung elastisch-plastischer Spannungswellen in einem unendlichen Medium betrachtet, welches einer idealen Spannungs-Verformungs-Kurve folgt, Trescas Fliesskriterium unterworfen ist und einen sphärischen Hohlraum enthält, wobei an der Fläche des Hohlraumes ein Stoss angenommen wird. Ein rechnerisches Verfahren, basiert auf endliche Differenzen, wird entwickelt and ein Beispiel gegeben.

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Notation radial stress - tangential stress - K yield stress - _rr non-dimensional radial stress ( /K) - non-dimensional tangential stress ( /K) - , Lame's constants - K _b Bulk constant (=(3+2)/3) - v Poisson's constant - Material density - C Elastic wave velocity (=((+2)/)^1/2) - C _p Plastic wave velocity (=(K _b /)^1/2) - distance from center of cavity - r ₀ cavity radius - v non-dimensional radial co-ordinate (= /r ₀) - time - t non-dimensional time (=C /r ₀) - radial displacement - u non-dimensional radial displacement (=/r ₀) - particle velocity - v non-dimensional particle velocity (= /C) - pressure - P(t) non-dimensional pressure (= /K)     

On vector-valued random Fourier series

Consider a (complex) Banach spaceX, such thatX C_O, and vectors(X _i ) _i ofX. Consider an independent standard normal sequence(g _i ) _i . Then if anX-valued random Fourier series _{|k| n} e ^ikt g _k x _k satisfies

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

The Halász Inequality for Additive Function of Rational Argument

We obtain an upper bound for the quantity . Here I is an interval, is the set of rational numbers q=m/n I such that nx, and f(q) is an arbitrary real-valued additive function of rational argument. The interval I and function f may depend on x3.     

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.     

Compound Poisson approximation for unbounded functions on a group,with application to large deviations

Summary A second order error bound is obtained for approximating h d by h d , where is a convolution of measures andQ a compound Poisson measure on a measurable abelian group, and the functionh is not necessarily bounded. This error bound is more refined than the usual total variation bound in the sense that it contains the functionh. The method used is inspired by Stein's method and hinges on bounding Radon-Nikodym derivatives related to . The approximation theorem is then applied to obtain a large deviation result on groups, which in turn is applied to multivariate Poisson approximation.Research of the second author was supported by Schweizerischer Nationalfonds     

On Universal Representation of Random Graphs

It is shown that every probability measure on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v₁, v₂, . . .} and a sequence of random graphs gn on vertices {v₁, . . . , v_n} such that . In particular, for Bernoulli graphs with stable property Q, can be strengthened to: probability space (, F, P), set of infinite graphs G(Q) , F with property Q such that .AMS Subject Classification: 05C80, 05C62.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.