On recurrence

Summary LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,) and letf: X be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={:f– is recurrent}. If , then R()={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Isotrope Kurventheorie aufS 2 × ℝ

In this note we define an isotropic metric on the threedimensional manifoldS ² × . This metric will allow an symmetric riemannian connection , wich will be used to do differential geometry on S² × . We develope theory of curves onS ² × and show some relations to the theory of curves of threedimensional isotropic spaceI ₃.     

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by . We show that the means of the critical index , do not mapL ^p,( ^N ) intoL ^p,( ^N ), but they map radial functions ofL ^p,( ^N ) intoL ^p,( ^N ). Moreover, iff is radial and in theL ^p,( ^N ) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N . We also observe that the means of the function|x| ^–N/p, which belongs toL ^p,( ^N ) but not to the closure of test functions, converge for nox.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

An algorithm for calculating one subgradient of a convex function of two variables

Summary We present an algorithm which enables us to calculate one particular subgradient of a convex functionf: ² at a given point. Such a calculation is required in many existing numerical methods for convex nondifferentiable optimization. The novelty of our approach lies in the assumption that only the values off are computable and no analytical formula for the subdifferential is known. We include some numerical examples.     

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

Summary We study here the discretisation of the nonlinear hyperbolic equationu _t +div(vf(u))=0 in ³ × ₊, with given initial conditionu(.,0)=u ₀(.) in ², wherev is a function from ² × ₊ to ² such that divv=0 andf is a given nondecreasing function from to . An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.     

Gewichtete Volumsmittelwerte von Simplices,welche zufällig in einem konvexen Körper des ℝ gewählt werden

For any set ofn+1 pointsx ₁, ...,x _n+1F we denote byv(C(x ₁,...,x _n+1)) then-dimensional oriented volume of the convex hullC(x ₁,...,x _n+1) of these points. With a fixed symmetric functionf: >> strictly monotone increasing on the nonnegative real line, we consider the real functional RODEL on the set of all convex bodiesK of ⁿ with absolute volume |v(K)|=1 and assert, that it takes its minimal value on the ellipsoids with absolute volume 1.     

Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken

LetP denote a polyhedral 2-manifold in ³, i.e. a 2-dimensional cell-complex in ³ whose underlying point-set is a closed connected 2-manifold. A vertexv ofP is called convex if at least one of the two components into whichP divides a sufficiently small ball centered atv is convex. It is shown that every polyhedral 2-manifold in ³ of genusg>–1 contains at least five non-convex vertices and that for every positive integerg this bound is attained, i.e. there exists a polyhedral 2-manifold in ³ of genusg with precisely five non-convex vertices.

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Mit 2 Abbildungen     

Some function classes closed under infimal convolution

It is shown that the following classes of functionsf, each defined on some subsetD of a fixed Hausdorff topological vector spaceE, are closed under the binary operation of infimal convolution: (a) the class of functionsf:D[–, ) having connectedstrict lower level sets; (b) the class of functionsf:D having compact connected lower level sets; and (c) the class of functionsf:D having compact lower level sets and for which every local minimizer is global. In (a),E need not be Hausdorff; while in (a) and (b), the word connected may be replaced by the word path-connected.     

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

275-01275-01275-01Regularity of the spatial derivatives of the solutions to parabolic systems in divergence form

Sunto Si dimostra che, nelle ipotesi: fⁱ L^2,(Q, ^N), 0<< n+2, i=0,1,..., n, uL^2,(Q,^N)H _–T ^*0,1/2() (Q,^N), D_iuL^2,(Q,^N),i=1,2,...,n, la soluzione v: Q ^N del problema di Cauchy-Dirichlet: ha derivate spaziali D_iv appartenenti a L^2,(Q,^N) e che sussiste la maggiorazione:.

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Lavoro eseguito con contributo finanziario del M.U.R.S.T. e nell'ambito del G.N.A.F.A. del C.N.R.     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

On Two-Point Boundary Conditions in Optimal Control Problems

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] ⁿ is continuous, u: [a,b] ^k-n is piecewise continuous and left continuous, h₀,h₁: ^{n q} are continuously differentiable, and g:[a,b]× ^{k n} is continuous. The paper finds functions _i C¹([a,b]× ⁿ ) such that (x(t),u(t)) is a solution of the governing equation if and only if

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .