Inequalities for moments of tails of random variables,with a queueing application

Summary For a non-negative random variable X and 1 such that EX<, E(X-Y) ⁺ /{E(X-y) ₊}⁺ is monotonic-decreasing in y, and hence no smaller than EX . Inequalities for E(X-Y) ₊ E(, 1, y, z0) are also given. This relation enables an inequality of Kingman for the mean waiting time in a stationary GI/G/1 queue to be sharpened.Work done as Visiting Fellow, Department of Statistics, University of Melbourne     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

Summary The Lyapunov exponents _{1 2}... _d for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting exponential growth rates of tangent vectors under the flow. This paper shows how the Lyapunov exponents are related to measure preserving properties of the stochastic flow on M and of the induced stochastic flow on the projective bundle PM. Relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow. The results include the following. If M is compact and if the one-point motion on M is a non-degenerate diffusion with stationary probability measure then ₁+...+ _d 0 with equality if and only if the flow preserves almost surely; if in addition the induced one-point motion on PM satisfies a weak non-degeneracy condition then ₁=...= _d if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely.     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains

Summary We show that ifD ⁿ ,n3,n3, is a bounded uniform domain, then the lifetime of the Doobh-paths inD for elliptic diffusions in divergence form is finite. This result holds for any bounded domainD in the plane.Research supported by a Bantrell Fellowship     

Indescribable cardinals without diamonds

Summary We show that form, n1 the existence of a _n ^m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a _n ^m indescribable cardinal together with the Generalized Continuum Hypothesis.     

Gauge theorems for resolvents with application to Markov processes

Summary LetV=(V )0 be a (not necessarily sub-Markovian) resolvent such that the kernelV for some 0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV ₀ is bounded.This result will be applied to resolventsU ^M arising from perturbation of sub-Markovian right resolventsU by multiplicative functionalsM (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao.     

Simplexes with Prescribed Edge Lengths in Minkowski and Banach Spaces

Existence of even a single regular d-simplex in an arbitrary Minkowski space M ^d of dimension d 4 is questionable. At least, for any d 4 there is an example of M ^d with four equidistant points in it which cannot be augmented by any fifth such point. At the same time, regular tetrahedra in M ^d , d 3, and regular triangles in M ^d , d 2, can be constructed as freely as in E ^d . Suppose that the Banach-Mazur distance between the unit balls of M ^d and E ^d satisfies

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

Sample path behaviour in connection with generalized arcsine laws

Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA ⁺=(A ⁺ (t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law.     

On the construction and calculation of optimal nonlifting critical airfoils

Numerical calculations are carried out in the hodograph plane to construct optimal critical airfoil shapes and the flow about them. These optimal airfoil shapes give the highest free-stream Mach numberM for a given thickness ratio and tail angle _t (nonlifting) for which the flow is nowhere supersonic. A relationship betweenM and for various _t is given. Analytical and numerical solutions to the same problem are found on the basis of transonic small-disturbance theory. These results provide a limiting case asM 1, 0 and agree well with the calculations of the full problem. Using a numerical method to calculate the flow about general (subsonic) airfoils, a comparison is made between the critical free-stream Mach numbers for some standard airfoil shapes and the optimal free stream Mach number of the corresponding and _t . A significant increase in the critical free-stream Mach number is found for the optimal airfoils.     

Some dimension results for super-Brownian motion

Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y _t)_t0 be a super-Brownian motion on ^d (d2) andH be a Borel subset of ^d . We determine the Hausdorff Dimension of {t0; SuppY _tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY _t as a function of the dimension ofB.     

A topological analog of Halmos' conjugacy lemma

Using a new lemma indicated by the title, and a recent measure preserving version of Lusin's Theorem, we prove the following theorem: Any isomorphism-invariant measure theoretic property which is typical for automorphisms of a Lebesgue space is also typical for Lebesgue measure preserving homeomorphisms of the unit cubeI ⁿ ,n2. We also prove a partial converse of this theorem. Taken together, these results clarify the relationship between pairs of theorems proved by several authors, which established the typicality of specific properties (such as ergodicity and weak mixing) separately in the measurable and continuous cases.     

On schematic orthogonal arraysOA(q t,k, q,t)

Am × k matrixA, with entries from a set ofq 2 elements, is called an orthogonal arrayOA(m, k, q, t) (t 2) if eachm × t submatrix ofA contains all possible 1 ×t row vectors with the same frequency(m = q ^t ). We call the array schematic if the set of rows ofA forms an association scheme with the relations determined by the Hamming distance. In this paper we determine the schematic orthogonal arraysOA(q ^t ,k, q, t) with2t – 1 > k.     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

Lifetime of conditioned Brownian motion in Lipschitz domains

Summary If D- ^d , d3, is bounded and has Lipschitz boundary then the expected lifetime of any Brownian h-path process in D is finite.     

Equilibrium time evolutions and BBGKY hierarchy equations of large classical systems

Summary We show the existence of a time evolution {P _t ; t of a locally perturbed equilibrium state P of infinitely many particles in {suv}, v1, evolving under the action of the infinite Newtonian dynamics associated to the same smooth, finite range pair potential as the equilibrium state itself. Moreover, it is shown that {P _t ; t solves the weak BBGKY hierarchy equations. The treatment of this problem will be done in the general setting of so called ( ) point processes developed in [11, 10] and [4] and will require the method of moments.     

Computations on the growth of the first factor for prime cyclotomic fields

Denote byh(p) the first factor of the class number of the prime cyclotomic fieldk(exp (2i/p)). The theorem:h(p ₂)>h(p ₁) if 641 p ₂>p ₁ 19 is proved by straightforward computation.     

On the long term behavior of some finite particle systems

Summary We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ ^d . Let _t denote this system, and let _t ^N denote a finite version of _t defined on the torus [–N,N] ^d Z ^d . Ford3 we prove that for stationary, shift ergodic initial measures with density , that ifT(N) andT(N)/(2N+1)^d s[0,] asN, then {v }, 0 is the set of extremal invariant measures for the infinite system _t andQ _s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg