When are random graphs connected

Several results concerning the connectivity of infinite random graphs are considered. A necessary sufficient condition for a zero–one law to hold is given when the edges are chosen independently. Some specific examples are treated including one where the vertex set isN and the probability that an edge joiningi toj is present depends only on |i−j|.     

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Fréchet spaces. Our main result is the following one: LetE be a Fréchet space which is neither Montel nor isomorphic to a closed subspace ofX × , withX a Banach space, also assume thatE can be written asFG withF andG infinite dimensional closed subspaces ofE not isomorphic to , thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Köthe echelon space of orderp, 1<p<, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.     

Critical densities for random quotients of hyperbolic groups

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.     

Jackknifed random weighting for Cox proportional hazards model

The Cox proportional hazards model is the most used statistical model in the analysis of survival time data.Recently,a random weighting method was proposed to approximate the distribution of the maximum partial likelihood estimate for the regression coefficient in the Cox model.This method was shown not as sensitive to heavy censoring as the bootstrap method in simulation studies but it may not be second-order accurate as was shown for the bootstrap approximation.In this paper,we propose an alternative random weighting method based on one-step linear jackknife pseudo values and prove the second accuracy of the proposed method.Monte Carlo simulations are also performed to evaluate the proposed method for fixed sample sizes.     

On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh-Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues.In the present paper, we derive upper bounds for proximity of the Ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace. The main result is that the absolute value of the perturbations in the Ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation. The constant is the spread in the matrix spectrum, i.e. the difference between the largest and the smallest eigenvalues of the matrix. It’s shown that the constant cannot be improved. We then generalize this result to arbitrary unitarily invariant norms, but we have to increase the constant by a factor of .Our results demonstrate, in particular, the stability of the Ritz values with respect to a perturbation in the trial subspace.     

Isoperimetric inequalities and random walks on quotients of graphs and buildings

Let be a Euclidean or hyperbolic building and let GAut be a locally compact unimodular group, which acts strongly transitively on . We use graphs , quasi-isometric to , to study asymptotic properties of quotients , where is a discrete subgroup of G. If G has Kazhdans property (T) we show that such quotients satisfy strong isoperimetric inequalities. This yields new examples of graphs with positive Cheeger constant. Such graphs cannot be bi-Lipschitz embedded into Hilbert space. Moreover, simple random walks on such quotients are shown to be recurrent if and only if is a uniform lattice in G.Mathematics Subject Classification (1991): 11E95, 22E40, 22E50, 51E24, 60G50in final form: 10 October 2003     

SeparableL 1 preduals are quotients ofC(Δ)

It is proved that every separable predual space of anL ₁ space is a quotient space ofC(Δ). Johnson's work was supported in part by NSF GP-33578.     

On subspaces of quotients of (ΣG n ) lp and (ΣG n ) co

Let 1<p<∞ (resp.p=∞). Then every ℒ _p -subspace of a quotient space ofl _p (resp.c ₀) is isomorphic tol _p (resp.c ₀). Supported by NSF GP-33578     

Triplewhist tournaments that are also Mendelsohn designs

We construct new families of whist tournaments that are at the same time both triplewhist tournaments and directedwhist tournaments. In particular, we construct such a design on v elements whenever v is a produce of primes p_i p_i ≥ 29, p_i ≡ 5 (mod 8). It follows that, for such v, two SOLSSOMs exist sharing the same mate. © John Wiley & Sons, Inc. J Combin Designs 5: 397–406, 1997     

When automorphism-coinvariant modules are quasi-projective

In this paper we provide conditions under which automorphism-coinvariant modules over a right perfect ring are quasi-projective.