Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

Extension of the Zorn lemma to general nontransitive binary relations

Let be an irreflexive (strict) binary relation on a nonempty setX. Denote the completion of by , i.e.,yx ifxy does not hold. An elementx ^* X is said to be a maximal element of onX ifx ^* x, xX. In this paper, an extension of the Zorn lemma to general nontrasitive binary relations (may lack antisymmetry) is established and is applied to prove existence of maximal elements for general nontrasitive (reflexive or irreflexive) binary relations on nonempty sets without assuming any topological conditions or linear structures. A necessary and sufficient condition has been also established to completely characterize the existence of maximal elements for general irreflexive nontrasitive binary relations. This is the first such result available in the literature to the best of our knowledge. Many recent known existence sults in the literature for vector optimization are shown to be special cases of our result.This work was supported in part by AFSOR Grant 91-0097.The author is grateful to the referees and Professor P. L. Yu for their comments and suggestions that led to this improved paper.     

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.

---

---

Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

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.     

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

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].     

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 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     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

The Darling-Erdős theorem for sums of I.I.D. random variables

Summary We obtain a general Darling-Erds type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erds theorem holds in its classical formulation if and only ifE[X ² 1 {|X|t}]=o((loglogt)^-1) ast. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.     

Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Minimax risk overl p -balls forl p -error

Summary Consider estimating the mean vector from dataN _n (, ² I) withl _q norm loss,q1, when is known to lie in ann-dimensionall _p ball,p(0, ). For largen, the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp. Obvious exceptions aside, the limiting ratio equals 1 only ifp=q=2. Our arguments are mostly indirect, involving a reduction to a univariate Bayes minimax problem. Whenp, simple non-linear co-ordinatewise threshold rules are asymptotically minimax at small signal-to-noise ratios, and within a bounded factor of asymptotic minimaxity in general. We also give asymptotic evaluations of the minimax linear risk. Our results are basic to a theory of estimation in Besov spaces using wavelet bases (to appear elsewhere).     

A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Summary A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (-valued) random variables {X _j , j1} and a kernel h: ^m H, m2, which is degenerate for the common distribution function of X _j , j1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=.     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.

     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

Ein Störungssatz für abgeschlossene,lineare Operatoren

Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6].     

Embedding (p,p−1) graphs in their complements

A graphG is embeddable in its complement ifG is isomorphic with a subgraph of . A complete characterization is given of those (p,p−1) graphs which are embeddable in their complements. In particular, letG be a (p,p−1) graph wherep≧6 ifp is even andp≧9 ifp is odd; thenG is embeddable in if and only ifG is neither the starK _1,p−1 norK _1,n ∪C ₃ withn≧4.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

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.