Zero product preserving maps on

The main result of the paper characterizes continuous bilinear maps from C¹[0,1]×C¹[0,1] into a Banach space X with the property that (f,g)=0 whenever fg=0. This is applied to the study of zero product preserving operators on C¹[0,1], and operators on C¹[0,1] satisfying a version of the condition of the locality of an operator.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Quadratic Covariation and Itô's Formula for Smooth Nondegenerate Martingales

In this paper we prove the existence of the quadratic covariation [f(X),X], where f is a locally square integrable function and X _t = ^t ₀ u _s dW _s is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus.     

Locally Compact Path Spaces

It is shown that the space X^[0,1], of continuous maps [0,1]X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T₁-space X, it follows that X^[0,1] is locally compact if and only if X is locally compact and totally path-disconnected. Mathematics Subject Classifications (2000) 54C35, 54E45, 55P35, 18B30, 18D15.     

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

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.

     

Absolutely continuous invariant measures forC 2 unimodal maps satisfying the Collet-Eckmann conditions

Summary In this paper we show that unimodal mappingsf[0, 1][0, 1] have absolutely continuous measures of positive entropy if these maps areC ² and satisfy the so-called Collet-Eckmann conditions. No conditions on the Schwarzian derivative off are assumed.     

Extension of stationary stochastic processes

Summary LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX ^{[0, 1]}, equipped with the Borel -field of the product space. We prove the existence of a stationary extension of this process to ₀ ⁺ . Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX ^{[0, 1]} is equipped with the product of the Borel -fields.Corporate Research and Development, SIEMENS AG, D-81730 Munich, Germany     

Sur les équations fonctionnelles aux itérées

Summary We study solutions of functional equationsP(f ^[10] ,,f ^[s] ) = 0, whereP is a non zero polynomial ins + 1 variables andf ^[k] denotes thekth iterate of a functionf. We deal with three distinct cases: first,f is an entire function of a complex variable, we show then thatf is a polynomial. Second, we also prove thatf is a polynomial if it is an entire function of ap-adic variable. Third, we considerf a formal power series with coefficients in a number fieldK; subject to some apparently natural restrictions onf and onP, we find thatf is an algebraic power series over the ring of polynomials inK[x].

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Sur les équations fonctionnelles aux itérées

     

Asymptotic behaviour of oscillatory solutions of n-th order differential equations with quasiderivatives

Sufficient conditions are given under which the sequence of the absolute values of all local extremes of y[i], i {0,1,..., n – 2} of solutions of a differential equation with quasiderivatives y ^[n] = f(t, y ^[0],..., y ^[n–1]) is increasing and tends to . The existence of proper, oscillatory and unbounded solutions is proved.     

Parallel interval multisplittings

Summary We introduce interval multisplittings to enclose the setS={A^–1b|A[A], b[b]}, where [A] denotes an interval matrix and [b] an interval vector. The resulting iterative multisplitting methods have a natural parallelism. We investigate these methods with respect to convergence, speed of convergence and quality of the resulting enclosure forS.Dedicated to the memory of Peter Henrici     

Effectively open real functions

A function f is continuous iff the pre-image f^-1[V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical open mapping theorems in analysis provide a variety of sufficient conditions for openness.By the main theorem of recursive analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping Vf^-1[V] being effective: given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f^-1[V]. Analogously, effective openness requires the mapping Uf[U] on open real subsets to be effective.The present work combines real analysis with algebraic topology and Tarski's quantifier elimination to effectivize classical open mapping theorems and to establish several rich classes of real functions as effectively open.     

Bisection is optimal

Summary We seek an approximation to a zero of a continuous functionf:[a,b] such thatf(a)0 andf(b)0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b–a)/2 ⁿ⁺¹, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.     

On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

On periodic points under the iteration of additive polynomials

Let F be a field and f(X) F[X]. An element P of F is called a (pre)periodic point of f if the sequence P, f(P), f(f(P)), is (eventually) periodic. In the case where F is a function field of characteristic p>0 and f(X)=aX ^q +bX or f(X)=aX ^{q 2}+bX ^q +cX with q a power of p, we try to give effective upper bounds (depending only on q and F) for the number of (pre)periodic points of f in F. Mathematics Subject Classification (2000):primary: 11C08, secondary: 11G09, 37E15     

Unification and extensions of some stability theorems in mathematical programming

In this paper, we provide the stability theorem for the program: inf{f(x, t)|xH(t)}, using the uniformlyN-type functions (also called -chainable functions^[10]). This theorem generalizes the results of Dantzig^[1], Hogan^[2], Greenberg^[3], Ying Mei-qian^[4] et al.Project supported by the Science Foundation of the Chinese Academy of Science.     

Approximate necessary conditions for locally weak Pareto optimality

Necessary conditions for a given pointx ⁰ to be a locally weak solution to the Pareto minimization problem of a vector-valued functionF=(f ₁,...,f _m ),F:XR ^m,XR ^m, are presented. As noted in Ref. 1, the classical necessary condition-conv {Df ₁(x ⁰)|i=1,...,m}T ^*(X, x ⁰) need not hold when the contingent coneT is used. We have proven, however, that a properly adjusted approximate version of this classical condition always holds. Strangely enough, the approximation form>2 must be weaker than form=2.The authors would like to thank the anonymous referee for the suggestions which led to an improved presentation of the paper.     

Outer approximation algorithm for nondifferentiable optimization problems

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ⁿ can be expressed as minimizing max{g(x, y)|y X}, whereg is a support function forf[f(x) g(x, y), for ally X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.This research was supported by the Science and Engineering Research Council, UK, and by the National Science Foundation under Grant No. ECS-79-13148.