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=.     

On a problem of Erdös and Alladi

We give an estimate for the quantity {f(n):nx, p(n)y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, ({(n):nx, p(n)y}) ({1:nx, p(n)y})^–1 tends to zero, asx, uniformly iny2, and thus settle a conjecture of Erdös.Supported by a grant from the Deutsche Forschungsgesellschaft.     

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     

Pinching constants for hyperbolic manifolds

Summary We show in this paper that for everyn4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK–1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with –1K–H.     

Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

Summary All rational approximations to exp(z) of order 2m– (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.     

On the structure of stationary flat processes. II

Summary The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure or a simple point process on the space of k-flats in R ^d is a.s. invariant or a Cox process respectively. The problems for and are related by the fact that is Cox whenever the Papangelou conditional intensity measure of (a thinning of) is a.s. invariant. In particular, is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure and has no (so called) outer degeneracies. When k=d–22, no absolute continuity is needed, provided that the first moments exist and that has no inner degeneracies either. Under a certain regularity condition on , it is further shown that and are simultaneously non-degenerate in either sense.     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

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].     

Characterization of allA-stable methods of order 2m-4

The characterization ofA-stable methods is often considered as a very difficult task (see e.g. [1]). In recent years, simple proofs have been found for methods of orderp2m-2 (see [2], [3], [7]). In this paper, we characterize theA-acceptable approximations of orderp 2m-4 and apply the result to 12-parameter families of implicit Runge-Kutta methods.     

Problèmes de prediction pour le processus de Wiener a deux parametres

Summary We give a decomposition of the two-parameter real Wiener process (W(x, y), (x, y)(⁺)² by using the Fourier transform defined on the group which leave it invariant. Then we use this decomposition to obtain some prediction results for this process, which allow us to give upper and lower bounds for the correlation coefficient of the vector-spaces respectively generated by the two families {W(x, y), xy s} and {W(x, y), xy t}, 0s t, {-1,1}.     

Existence of quadrature surfaces for positive measures with finite support

Let be a positive measure with finite support in ⁿ (n2). Then we show that there is a bounded open set , containing the support of , whose single-layer potential coincides with the potential of outside .     

Metrics with finite sets of primitive extensions

In this paper, we give a complete characterization for the class of rational finite metrics with the property that the set () of primitive extensions of is finite. Here, for a metric on a setT, a positive extensionm of to a setV T is calledprimitive if none of the convex combinations of other extensions of toV is less than or equal tom. Our main theorem asserts that the following the properties are equivalent: (i) () is finite; (ii) Up to an integer factor, is a submetric of the path metric d ^H of a graphH with |(d ^H )=1; (iii) A certain bipartite graph associated with contains neither isometrick-cycles withk6 nor induced subgraphsK _3,3 ^– . We then show that () is finite if and only if the dimension of the tight span of is at most two. We also present other results, discuss applications to multicommodity flows, and raise open problems.This research was supported by grant 97-01-00115 from the Russian Foundation of Basic Research and a grant from the Sonderforschungsbereich 343, Bielefeld Universität, Bielefeld, Germany.     

The self-similar solutions to a fast diffusion equation

A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R ⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.     

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.     

d≤1 ∪d≥25 and constrained KP hierarchy from BRST invariance in thec≠3 topological algebra

The BRST invariance condition in a highest-weight representation of the topological ( twisted N=2) algebra captures the invariant content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the Kontsevich-Miwa-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into c=–26 reparametrization ghosts + matter + Liouville, while a similar splitting involving c=–2 ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the Liouville differ also by their ghost numbers, which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c3 provides a two-fold covering of the allowed regiond1 d25 of the matter central charge d via d=(c+1)(c+6)/(c–3). The Liouville field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 239–250, May, 1993.     

Remark on the Helly number for strongly convex sets on Riemannian manifolds

Replacing convex by strongly convex we show that Helly's famous intersection theorem holds on every Riemannian n-manifold in the following form: The intersection of k relatively compact, strongly convex subsets of M (kn+i2) is nonvoid as soon as any n+i of these sets have a nonvoid intersection, where i=2 if M is homeomorphic to the standard n-sphere and i=1 otherwise.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

Criterion for the compactness of a class of functions of W m l (Ω), (1 <mι ≤ n)

A criterion is established for the compactness of a class of functions u(x) W _m (), (1 <m n) generalizing Morrey's criterion.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 733–741, June, 1970.     

Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems

Summary This paper deals with polynomial approximations ø(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument, denoted by –R(ø), at which ø is absolutely monotonic. For given integersp1,m1 we determine the maximum ofR(ø) when ø varies over the class of all polynomials of a degree m with (forx0).     

On a transformation operator for a system of Sturm-Liouville equations

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation^–y + My=²y (M is a constant Hermitian matrix) into a solution of the matrix equation^–y+Q(x)y+My=²y (the matrix function Q(x) is continuously differentiable for 0 x< and it is Hermitian for each x belonging to [0, )); we study some properties of the kernel of the transformation operator.Translated from Matematicheskii Zametki, Vol. 11, No. 5, pp. 559–567, May, 1972.The authors express their thanks to B. M. Levitan for a discussion.