Tournaments with prescribed regular automorphism group

A tournamentX is a TRR for a groupG if (a)G acts regularly on the vertices ofX and (b) Aut(X) is isomorphic toG. We correct some previous work of Babai and Imrich by showing thatZ ₂ ³ andZ ₃ ³ are the only groups of odd order without TRR's. Our methods are perhaps of independent interest, since we use a probabilistic approach.     

Automorphisms inc 0,l 1 andm

It is proved that ifX=c ₀ orm and ifY andZ are subspaces ofX withX/Y andX/Z non-reflexive, then any isomorphism ofY ontoZ has an extension to an automorphism ofX. A dual result is obtained forX=l ₁. This research was partially supported by NSF-GP-8964.     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

On a characterization of probability distributions on locally compact abelian groups

Summary In one of his recent papers, I. J.Kotlarski has proved the following result. IfX ₁,X ₂,X ₃ are three independent real random variables and if the characteristic function of the pair (Z ₁,Z ₂) whereZ ₁=X ₁-X ₂,Z ₂=X ₁-X ₃ does not vanish, then the distribution of (Z ₁,Z ₂) determines the distributions ofX ₁,X ₂,X ₃ up to a change of location. He extended this result to random variables taking values in a Hilbert space in another paper. Our aim in this paper is to extend the above result to probability distributions on locally compact abelian groups.     

On some extremal problems in graph theory

The author proves that ifC is a sufficiently large constant then every graph ofn vertices and [Cn ^3/2] edges contains a hexagonX ₁,X ₂,X ₃,X ₄,X ₅,X ₆ and a seventh vertexY joined toX ₁,X ₃ andX ₅. The problem is left open whether our graph contains the edges of a cube, (i.e. an eight vertexZ joined toX ₂,X ₄ andX ₆).     

Weakly compact subsets ofL 1(μ,X) andbvca(Σ,X)

In this paper we study the weakly compact subset ofL ₁(μ,X) and bvca(Σ,X), supposing that proper subsets ofX andX ^* have the Radom-Nikodym property.     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

On the local minimaxity of a test of independence in incomplete samples

Summary LetX=(X ₁,X ₂) be ap-dimensional normal variable. Suppose that we haven independent observations onX andm, independent observations onX _i ,i=1,2, respectively. For testing the independence betweenX ₁ andX ₂, Eaton and Kariya (1983),Ann. Statist.,11, 654–665) obtains a locally most powerful test among the class of tests invariant under the group of affine transformations. In this paper, this LMPI test is shown to be locally minimax within the class of all tests of the same level.     

On the number of antipodal or strictly antipodal pairs of points in finite subsets ofR d,II

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal pairs of points in a subsetX ofR ^d , which has cardinalityn. The pointsx _i, x_jX are called antipodal if each of them is contained in one of two different parallel supporting hyperplanes of the convex hull ofX. If such hyperplanes contain no further point ofX, thenx _i, x_j are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for givend andn. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of speciald-simplices, and finally a generalization (concerning strictly antipodal segments) is proved.Research (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1817     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Some results on the residual life at random time

In this paper, we consider the residual life at random time, i.e.X _Y =X−Y\X>Y, whereX andY are non-negative random variables. We establish a number of stochastic comparison properties forX _Y under various assumptions ofX andY. Under the assumption thatY has decreasing reverse hazard rate (DRHR), we show that ifX is in any one of the classes IFR, DFR, DMRL or IMRL thenX _Y is in the same class asX. We also obtain some useful bounds for the distribution and the moment ofX _Y . Because the idle time in classicalGI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system. This work is supported by the National Natural Science Foundation of China.     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

Multiple recurrence theorem for nilpotent group actions

We prove that ifX is a compact space andT ₁, ...,T _t are homeomorphisms ofX generating a nilpotent group, then there existx X andn ₁,n ₂, ... N such that for eachj=1, ...t.Supported by British Technion Society.     

Nonparametric bayesian estimation of a survival curve with dependent censoring mechanism

Summary Let (X ₁,Y ₁),..., (X _N ,Y _N ) be a random sample from a bivariate distribution functionF. Based on observing only (δ _i ,Z _i ) whereδ _i =1 ifX _i ≦X _i and =0 otherwise andZ _i =min{X _i ,Y _i } fori=1,...,n, we obtain the Bayes estimator ofF whenF is a Dirichlet process under the usual integrated squared error loss function. It should be pointed out here thatX _i andY _i neednot be independent which is the usual assumption in survival analysis models. The effect of this dependence can be seen clearly in the estimators obtained and also in the given example which illustrates the estimator when Freunds' bivariate exponential distribution is taken as the parameter of the Dirichlet process. The research of this author is supported in part by an NSF Grant MCS80-03244. The research of this author is supported by the NIH Grant NO: 1 R01 GM 28405.     

Some characterization results based on the conditional expectation of a function of non-adjacent order statistic (record value)

In this paper, we attempt to characterize a distribution by means ofE[ψ(X _k +s:n)|X _k:n =z]g(z), under some mild conditions on ψ(·) andg(·). An explicit result is provided in the case ofs=1 and a uniqueness result is proved in the case ofs=2. For the general case, an expression is provided for the conditional expectation. Similar results are proved for the record values, both in the continuous as well as in the discrete case (weak records).     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case

Summary LetX ₁,X ₂, ...,X _r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX _i =(X _i1 , ...,X _in ) and the random vectorY=(Y ₁, ...,Y _n ), their maximum, is defined byY _j =max{X _ij :1ir}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ ₁,Z ₂, ...,Z _s . It is then shown in this paper that the distributions of theZ _i 's are simply a rearrangement of those of theZ _j 's (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.     

Code length between Markov processes

LetX ₁ andX ₂ be two mixing Markov shifts over finite alphabet. If the entropy ofX ₁ is strictly larger than the entropy ofX ₂, then there exists a finitary homomorphism ϕ:X ₁→X ₂ such that the code length is anL ^p random variable for allp<4/3. In particular, the expected length of the code ϕ is finite. Research supported by KBN grant 2 P03A 039 15 1998–2001.     

The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).