On the consistency of conditional maximum likelihood estimators

Let {P _, : , H} be a family of probability measures admitting a sufficient statistic for the nuisance parameter . The paper presents conditions for consistency of (asymptotic) conditional maximum likelihood estimators for . An application to the Rasch-model (a stochastic model for psychological tests) yields a condition on the sequence of nuisance parameters which is sufficient for strong consistency of conditional maximum likelihood estimators, and necessary for the existence of any weakly consistent estimator-sequence.     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

Symmetriegerechte Basisvektoren für Permutationsdarstellungen aller endlichen Punktgruppen der Dimensionen 3 und 2

Zusammenfassung Es seiG eine endliche Untergruppe der orthogonalen Gruppe (det=±1) des ^k mitk=2 oder 3 undN eine endliche Menge von Punkten des ^k , welche unterG invariant ist. Dies gibt Anlass zu einer Permutationsdarstellung vonG im Vektorraum der komplexen Funktionen aufN.In Abschn. 3 wird für eine symmetriegerechte Basis angegeben. Dabei sind die Funktionswerte jeweils exakt tabelliert.

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Let G be a finite subgroup of the orthogonal group (det=±1) of ^k wherek=2 or 3 and letN be a finite set of points of ^k , which is invariant underG. In this way one gets a permutation representation ofG in the vector space of the complex functions onN.In Section 3, a symmetry adapted basis is given for , where the function values are tabulated exactly.

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Im Buch [1] wurden lediglich die Diedergruppen behandelt (in Abschn. 3.1).     

Randomized graph products, chromatic numbers, and the Lovász ϑ-function

For a graphG, let (G) denote the size of the largest independent set inG, and let (G) denote the Lovász -function onG. We prove that for somec>0, there exists an infinite family of graphs such that \alpha (G)n/2^{c\sqrt {\log n} }$$ " align="middle" border="0"> , wheren denotes the number of vertices in a graph. this disproves a known conjecture regarding the function.As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest.Incumbent of the Joseph and Celia Reskin Career Development Chair. Yigal Alon Fellow     

On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

In this paper, we prove the existence of an element of the group algebra A=F of a free groupF with two generatorsx andy over the field of complex numbersC such that, for any complexa andb for which ¦a¦=¦b¦=1, we haveA _a,b ()A=0, where _a,b ( is an automorphism ofA that mapsx,y intoax, by, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from Kourovka Notebook.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No.4, pp. 571–572, April, 1995.     

On a generalization of Morisita's model for estimating the habitat preference

The Morisita's model for estimating the habitat preference by the ant lionsGenuroides japonicus is generalized by introducing, in addition to the environmental densitiesa andb, a repulsivity parameter . The probability function of the numberL _n of individuals choosing fine sand to settle when a total ofn ant lions are introduced is examined. A heuristic and the minimum chi-square methods for estimating the parametersa, b and are discussed.     

Fuzzy (Θ, S)-continuity and fuzzy (Θ, S)-closed graphs

The concept of (,s)-continuity [6] is considered and studied in fuzzy setting. It is seen that althought it is independent with each of the concepts of fuzzy continuity [2], fuzzy -continuity [10], fuzzy almost continuity [1] and fuzzy semicontinuity [1]; it implies fuzzy weak continuity [1], but the converse may not be true. The image of a compact fts [2] under a fuzzy (,s)-continuous surjective function isS-closed [5]. Finally the concepts of fuzzy (,s)-closed graphs, fuzzy (,s)-T ₂ spaces and fuzzy Urysohn spaces are introduced and mainly their connections with fuzzy (,s)-continuity are studied.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

Spectral Characterizations of the Lovász Number and the Delsarte Number of a Graph

This paper gives spectral characterizations of two closely related graph functions: the Lovász number and a generalization ¹ of Delsarte's linear programming bound. There are many known characterizations of the Lovász number , and each one corresponds to a similar characterization of ¹ obtained by extremizing over a larger or smaller class of objects.The spectral characterizations of and ¹ given here involve the largest eigenvalue of a type of weighted Laplacian that Fan Chung introduced.     

Two poset polytopes

Two convex polytopes, called theorder polytope (P) andchain polytope (P), are associated with a finite posetP. There is a close interplay between the combinatorial structure ofP and the geometric structure of (P). For instance, the order polynomial (P, m) ofP and Ehrhart polynomiali((P),m) of (P) are related by (P, m+1)=i((P),m). A transfer map then allows us to transfer properties of (P) to (P). In particular, we transfer known inequalities involving linear extensions ofP to some new inequalities.Partially supported by NSF Grant No. 8104855-MCS and by a Guggenheim Fellowship.     

Eine ErweiterungT(V′) des Ordinalzahlensystems 58-0158-0158-01(Λ0) von G. Jäger

Summary This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V) is provable in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V).     

Quantitative measure of the amount of information acquired in a learning process

When Shannon presents the formula for the entropy of a memoryless source, he presupposes that the prior probabilities of the different source symbols are known. This paper deals with the quantity of information acquired when the prior probabilities of a binary source are learned from a sequence ofN source symbols or Bernoulli trials. Two learning methods are considered: Maximum likelihood estimation of a parameter by calculation of the relative frequency; and calculation of the posterior probability density for . For both methods the acquired information behaves as 1/2 logN + const. for largeN.     

Paradoxical subsets of hyperbolic and spherical discs

Given a disc D of radius r in H ² (resp. S ²) with <r (resp. <r), we determine the pairs (m,n) for which there is an (m,n)-paradoxical subset of D but not an (m–1, n)-paradoxical subset of D or an (m, n–1)-paradoxical subset of D.     

Error estimates over infinite intervals of some discretizations of evolution equations

Classical discretization error estimates for systems of ordinary differential equations contain a factor exp (Lt), whereL is the Lipschitz constant. For strongly monotone operators, however, one may prove that for a-method, 0<<1/2, the errors are bounded uniformly in time and with errorO(t)², if=1/2–|O(t)|. This was done by this author (1977), for an operator in a reflexive Banach space and includes the case of systems of differential equations as a special case.In the present paper we restate this result as it may have been overlooked and consider also the monotone (inclusive of the conservative) and unbounded cases. We also discuss cases where the truncation errors are bounded by a constant independent of the stiffness of the problem. This extends previous results in [6] and [7]. Finally we discuss a boundary value technique in the context above.Dedicated to Germund Dahlquist: a stimulating teacher and researcher     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

Existence of a universal attractor for a fully hyperbolic phase-field system

Here we study a nonlinear hyperbolic integrodifferential system which was proposed by H.G. Rotstein et al. to describe certain peculiar phase transition phenomena. This system governs the evolution of the (relative) temperature and the order parameter (or phase-field) . We first consider an initial and boundary value problem associated with the system and we frame it in a history space setting. This is done by introducing two additional variables accounting for the histories of and . Then we show that the reformulated problem generates a dissipative dynamical system in a suitable infinite-dimensional phase space. Finally, we prove the existence of a universal attractor.     

Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995