On the topology of foliations with a first integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle generates the first homology group of a generic fiber of . In particular, we will prove that for any two Lefschetz vanishing cycles ₀ and ₁ in a regular compact fiber of , there exists a mondromyh such thath( ₀)=± ₁.Partially supported by CNPq-Brazil.     

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.     

Une propriété élementaire des fonctions dont le laplacien est constant,en particulier des fonctions harmoniques

Résumé On montre que la valeur moyenne d'une fonction harmoniqueu sur le contour d'un polygone régulier est égale à sa valeur moyenne sur les rayons du polygone (segments joignant les sommets au centre). Plus généralement si u=c, alors , oùR est le rayon du cercle inscrit.

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Summary We show that the mean value of any harmonic functionu on the boundary of a regular polygon is equal to its mean value on the radii of the polygon (segments joining all vertices to the center). More generally, if u=c, then , whereR is the radius of the inscribed circle.

     

On the Characterisation of AG(n,q) by its Parameters asa Nearly Triply Regular Design

We show that a non-symmetric nearly triply regular designD with and in which every line has at least q points is AG(n,q) for prime power q > 2 and positiveinteger n 3.     

Kleine Nullstellen quadratischer Formen in Funktionenkörpern

Letk be a field of characteristic different from 2 andt an indeterminate overk. Let0 be a quadratic form inn variables with coefficients _ij = _ji ink[t]. We show that if vanishes on ad-dimensional subspace ofk(t) ⁿ , then there is a zero (x ₁ ,...,x _n )k[t] ⁿ –{(0,...,0)} with max max{deg _ij }. We also show, that the factor is best possible.

     

On the Center of the Generalized Liénard System

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)₁

Concentration phenomena in weakly coupled elliptic systems with critical growth*

In this paper we consider the weakly coupled elliptic system with critical growth

The Lie algebra of polynomial vector fields and the Jacobian conjecture

The Jacobian conjecture for polynomial maps :K ⁿ K ⁿ is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra of formal vector fields inn variables. To be precise, let be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of ,H the root spaces corresponding to linear forms onH and . Then every polynomial map :K ⁿ K ⁿ with invertible Jacobian matrix is an automorphism if and only if every automorphism of with (A) satisfies (A)=A.     

Accuracy of Computed Eigenvectors Via Optimizing a Rayleigh Quotient

This paper establishes converses to the well-known result: for any vector such that the sine of the angle sin(u, )=O(), we have

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set there exists a homeomorphism of the closed interval such that for an arbitrary function f the Fourier series of the function F(x,y) = f((x),(y)) converges uniformly on simultaneously over rectangles, over spheres, and over triangles.     

Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities

Extension of a class of fibered loops to kinematic spaces

The notion of the split extension of a commutative kinematic space is extended to the case of a weak K-loop with an incidence fibration (F, +, ). Theorem 1 states conditions under wich the quasi-direct productG F⁺ _Q with Aut(F, +) can be turned in a fibered incidence group (G, , o) such that (F, +, ) becomes embeddable inG, and Theorem 2 the additional assumption such that (G, , o) is even a kinematic space. In section 4, Theorem 3 shows that there are suitable examples of proper K-loops with an incidence fibration (derived from hyperbolic planes) on which one can apply Theorem 2.Dedicated to Erich Ellers on the occasion of his 70th birthdayResearch supported by M.U.R.S.T. 40% and by C.N.R. (G.N.S.A.G.A.)     

Comments on the Trotter product formula error-bound estimates for nonself-adjoint semigroups

LetA be a positive self-adjoint operator and letB be anm-accretive operator which isA-small with a relative bound less than one. LetH=A+B, thenH is well-defined on dom(H)=dom(A) andm-accretive. IfB is a strictlym-accretive operator obeying

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

Type,infratype and the Elton-Pajor theorem

Summary Consider a sequence(x _i ) _in of norm one vectors in a Banach space. For a subsetJ of {1,...,n} consider the equivalence constant (J) between(x _i ) _iJ and the ₁ basis, and consider (k)=min{(J); cardJ=k}. We give a near optimal relationship between the rate of decay of (k) and the averageE of over all choices of signs. In particular, we show that one can choosek such that, for some universal constantK, kE ² /Kn and . This is optimal within the logarithmic term. We also prove, that forp<2, the notions of type and infratype coincide.Oblatum 27-III-1991Work partially supported by an NSF grant     

Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

Let be the linear space of the Laurent polynomials and suppose that <, < is a positive-definite Hermitian inner product in with the additional property that . Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <, <, we derive Laurent–Jacobi matrices and for the multiplication operator and its inverse in . These matrices are real and symmetric, and generates a symmetric operator in the Hilbert space ² with natural basis { e _n } _{n = 0} . We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in ² has simple spectrum with generating vector e ₀. Let E be the spectral measure of A. Then the measure _{e 0} given by _{e 0}() =<E() e ₀, e ₀< for all Borel sets in , satisfies forf,g. In this way, we obtain a solution _{e 0} of the Strong Hamburger Moment Problem (SHMP) for which is dense in L ²( _{e 0}). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).     

A functional limit theorem for Erdös and Rényi's law of large numbers

Summary We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W _{N, n} } defined by where , {X _i : _i 1} is an iid sequence of random variables, andh(N)=[clogN] is relatively compact; the limit set is given by the set [xI ^*(x)1/c] whereI ^*(x) = ₀ ¹ I(x(t))dt andI is Cramér's rate function.     

On the metric theory of the nearest integer continued fraction expansion

Supposek _n denotes either (n) or (p _n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let denote the sequence of convergents to a real numberx and letc _n(x)) _n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants _n(x)) _n=1 by

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday