On the angular derivative and univalence

f . , , — , A f f(). , , f() 0 . , , ,A , f . , f() - f() . , , . (1976) ( ¦f(z)¦<1) . . (1969) ( ).     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

Stability of Jackson Type Network Output

We consider an open Jackson type queueing network N with input epochs sequence I={T _n ⁽⁰⁾,n0}, T ₀ ⁽⁰⁾=0, assume another input ={ _n ⁽⁰⁾} and denote _k =| _k ⁽⁰⁾–T _k ⁽⁰⁾|, ₀=0, _n =max_{1kn k} , n1. Let {T _n } and { _n } be the output points in network N and in modified network, with input , accordingly. We study the long-run stability of the network output, establishing two-sided bounds for output perturbation via input perturbation. In particular, we obtain conditions that imply max _kn |T _k – _k |=o(n ^1/r ) with probability 1 as n for some r>0. This result is also extended to continuous time. We consider successively separate station (service node), tandem and feedforward networks. Then we extend stability analysis to general (feedback) networks and show that in our setting these networks can be reduced to feedforward ones. Similar stability results are also obtained in terms of the number of departures. Application to a tandem network with the overloaded stations is considered.     

An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), ₂, ..., _n. Fiedler and others have shown that \det( I -A) ⁿ - ⁿ, for all > with equality for any such if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If ₁ = (A), ₂,...,_k are (all) the nonzero eigenvalues of A, then . We prove this conjecture for the case when the spectrum of A is real.     

On π-Solvable and Locally π-Solvable Groups with Factorization

We prove that, in a locally -solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow - and p-subgroups A , A _p and B , B _p , p , of the subgroups A and B, respectively, such that A B is a Sylow -subgroup of the group G and, for an arbitrary nonempty set ,

Sur le problème des marges

Summary LetX be a set,A an algebra of subsets ofX, m andM two mappings fromA to . Then there exists a finitely additive measure onA such thatmM if and only if for all the sequences (A ₁, ...,A _p ) and (B ₁, ...,B _q ) inA such that the inequality is satisfied. This simple condition permits us to deduce and generalize many previous results relating to the marginal problem.     

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

On the integersn for which Ω(n)=k (II)

Let (n) be the number of prime divisors ofn, counted with multiplicity. We denote byS(x, k) the set of thenx for which (n)=k, and byV _p(n) the exponent of the primep in the factorization ofn. In a previous paper we proved a result which implies that, ify=x/2 ^k tends to infinity withk>2loglogx where >1, then the distribution of the numbers on the setS(x, k) converges to the normal distribution of Gauss. Here, besides a slight improvement of that result, we give, for the moment of orderq of the above mentioned distribution, a formula which holds uniformly for 2loglogxklog (x/3)/log2 where 1<<3/2.     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

On the parallel factorization of matrices over principal ideal rings

We describe all the factorizations A=BC (up to associates) of a matrix A over a commutative principal ideal domain parallel to the factorization D^A= of its canonical diagonal form D^A ( and are diagonal matrices), that is, the factorizations such that the matrices B and C are equivalent to the matrices and respectively.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 96–100.     

On the stochastic independence properties of hard-core distributions

A probability measurep on the set of matchings in a graph (or, more generally 2-bounded hypergraph) ishard-core if for some : [0,), the probabilityp(M) ofM is proportional to . We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP () the matching polytope of , and >0, if the vector ofmarginals, (Pr(AM):A an edge of ), is in (1–) MP (), then the weights (A) are bounded by someA(). This eventually implies, for example, that under the same assumption, with fixed, as the distance betweenA, B tends to infinity.Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]–[13]).Supported in part by NSFThis work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.     

Looking for Selectors of Star Bodies

Let be a family of star bodies in R ⁿ (compact bodies in R ⁿ with nonempty kernels). A function s: R ⁿ is a selector for provided that s(A) ker A for every A . To every star body A and every : [0,) [0,) we assign a function _A: ker A R defined in terms of the radial function of translates of A. We prove that if A is convex and is concave and strictly increasing, then _A has a unique maximizer, which is referred to as the radial center of A induced by (Theorem 3.1). We extend the radial center map over some family of star bodies (Theorem 4.2). Further, we define a suitable metric, _st , the star metric, on the family of all the compact star sets in R ⁿ . This new metric is topologically stronger than the Hausdorff metric (Theorem 5.7). We study the continuity of our selectors with respect to _st .     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

Radon—Nikodym theorems for multimeasures and transition multimeasures

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Research supported by N. S. F. Grant DMS-8802688.     

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (, ) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d() such that

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Dualities for the Domany–Kinzel Model

We study the Domany–Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p ₁,p ₂)[0,1]². When p ₁= and p ₂=(2– ²) with (,)[0,1]², the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities of a site being open and of a bond being open. This paper treats dualities for the Domany–Kinzel model _t ^A and the DKdual _t ^A starting from A. We prove that , as long as one of A,B is finite and p ₂p ₁.     

Adjoints of slant Toeplitz operators II

Let be the unit circle {z|z|=1} and _n c _n e ⁱⁿ be a bounded measurable function on . Theslant Toeplitz operator A onL ² ( ) is defined by A e _n ,e _m =c _2m–n for allm, n wheree _n (z)=z ⁿ , . In this paper, we continue the study initiated in [6] onA ^* , the adjoint ofA . Specifically, we will show that for a certain dense set of continuous functions on ,A ^* is similar to some constant multiple of either a shift, or a shift plus a rank one operator.     

Nilpotent shifts on manifolds

On the lattice of manifolds of all algebras L we study the operator of nilpotent closure , where is a nilpotent manifold of -algebras. With a given system of identities defining, we construct a system ^*, giving the manifold It is proved that if does not contain , then the lattice of submanifolds of is the double of the lattice of submanifolds of. We describe the free and subdirect indecomposable manifolds of algebras . Let and A be adense retract of B. We denote by (B) the lattice of congruences on B. The theorem is proved: (B) is a complemented lattice if and only if (A) is a complemented lattice.Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 703–712, November, 1973.     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.