Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I ²(K) and that the induced natural homomorphism from K ₂ Sp(K)I ²(K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, -x 1, –y.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Nonlinear programming and an optimization problem on infinitely differentiable functions

A nonnegative, infinitely differentiable function ø defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ₀ ¹ ø(t)dt=1. In this article the following problem is considered. Determine _k =inf ₀ ¹ vø^(k)(t)dt, k=1,..., where ø^(k) denotes thekth derivative of ø and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining _k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k–1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., _k =k~2^2k–1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.This research was supported in part by the National Science Foundation under Grant No. GK-32712.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum'

Let M be a compact connected manifold of dimension n endowed witha conformal class C ofRiemannian metrics of volume one. For any integer k 0, we consider the conformal invariant _k ^c (C) defined as the supremum of the k-th eigenvalue _k (g) of the Laplace–Beltrami operator _g , where g runs over C.First, we give a sharp universal lower bound for _k ^c (C) extending to all k a result obtained by Friedlander andNadirashvili for k = 1. Then, we show that the sequence \{ _k ^c (C)\}, that we call `conformal spectrum',is strictly increasing and satisfies, k 0, _k+1 ^c (C) ^n/2– _k ^c (C) ^n/2 n ^n/2 _n , where _n is the volume of the n-dimensionalstandard sphere.When M is an orientable surface of genus , we also considerthe supremum _k ^top()of _k (g) over theset of all the area one Riemannian metrics on M, and study thebehavior of _k ^top() in terms of .     

Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus

Summary Two previous papers (in Vol. V) describe theory and some applications of the quotient-difference (=QD-) algorithm. Here we give an extension which allows the determination of the eigenvectors of a matrix. Letx ₍₀₎ ¹ , ...,x ₍₀₎ ⁿ be a coordinate system in whichA has Jacobi form (such a system may be constructed with methods ofC. Lanczos orW. Givens). Then the QD-algorithm allows the construction of a sequence of coordinate systemsx ₍₂₎ ¹ , ...,x ₍₂₎ ⁿ , (=0, 1, 2, ...) which converge for to the system of the eigenvectors ofA.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

Formule d'Itô pour des Diffusions Uniformément Elliptiques,et Processus de Dirichlet

Soit X une diffusion uniformément elliptique sur R ^d ,F une fonction dans H _loc ¹(R ^d ) et la loi initiale de la diffusion. On montre que si l'intégrale |F|²(x)U(x)dx est finie, oùU désigne le potentiel de la mesure , alors F(X) est un processus de Dirichlet. Si de plus, F appartient àH ² _loc(R ^d ) et si les intégrales |F|²(x)U(x)dx et |f _k |²(x)U(x)dx sont finies, pour les dérivées faibles f _k de F, alors on peut écrire une formule d'Itô. En particulier, on définit l'intégrale progressive F(X)dX et on prouve l'existence des covariations quadratiques [f _k (X),X ^k ].     

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

A conjecture on semi-field planes. II

Conclusion The results in the previous sections lend strong support to the conjecture made in the Introduction. Furthermore, if the long-standing conjecture concerning the solvability of autotopism groups for semi-field planes is correct then the probability of our conjecture being true is greatly increased. In any case the existence of a semi-field plane for which u() = 2, 3, or 4 would provide a counterexample to the earlier conjecture.There are examples of semi-field planes with u() = %. As mentioned in Example 2 of Section 3, one of the semi-field planes of order 16 has u()-5. For that plane, the five orbits of the autotopism group G in (G) have lengths 27, 36, 54, 54, 54. The union of the orbit having length 36 and one of those having 54 is the union of the points in (G) on 6 lines through a vertex U and the union of the remaining three orbits consists of the 135 points on the remaining 9 lines through U. There are also non-Desarguesian A-planes in which u() = 5; the semi-field plane of order 3⁴ coordinatized by the twisted field of Albert has u() = 5.Supported in part by NSF Grants No. MPS 75-05260 and MPS 76-06661     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Weak Sequential Completeness of β-duals of Double Sequence Spaces

We consider dual pairs E,E ⁽⁾ of double sequence spaces E and E ⁽⁾, where E ⁽⁾ is the -dual space of E with respect to the -convergence of double sequences for = p (Pringsheim convergence), bp (bounded p-convergence) and r (regular convergence). Motivated by Boos, Fleming and Leiger [3], we introduce two oscillating properties (signed P_OSCP(k), k {1,2}) for a double sequence space E such that the signed P_OSCP(1) guarantees the (E ^(p), E)-sequential completeness of E ^(p), whereas the signed P_OSCP(2) implies the equalities E ^(r) = E ^(bp) = E ^(p) and the (E ⁽⁾, E)-sequentialcompleteness of E ⁽⁾ for = bp and r.     

On the asymptotic behaviour of sums Σg(n){x/n} k

In this paper we estimate the difference between the sum given in the title (whereg(t) is an arbitrary real-valued non-decreasing function,k is a positive integer and summation is extended over all positive integersnx) and the corresponding integral, obtaining the boundO(g(x)x ^1/3logx). Furthermore, we show that these differences (for giveng and varyingk) are all approximately equal, with an error term ofO(g(x)x ^3/10). Finally it is remarked without proof that these estimates can be refined toO(g(x)x ) (with any >0,329...) resp.O(g(x)x ^109/382).     

Two-parameter Vilenkin multipliers and a square function

Two-parameter Vilenkin systems will be investigated. First we give a general sufficient condition for multipliers to be bounded between two-dimensional Hardy spaces H ^q(0<q1). By means of interpolation and duality argument, this theorem can be extended to other spaces. As a consequence, we can prove the (H ^q , L ^q)-boundedness of the Sunouchi operator U with respect to two-parameter Vilenkin systems for all 0 <q 1. Moreover, the equivalence f{H_q} ~ Uf_q (f H^q)follows for 1/2<q 1.     

Über eine Verallgemeinerung des Plateauschen Problems

Let T be the domain in ^N defined by the inequalities O < ₁ < ... < _N < +. Put _N+k = /2(1+k) (k=1,2,3), _N+4=₁+2, and denote byF() the set of functions x=x(u,v)=(x¹(u,v),...,x^p(u,v)), (p2) of class , where B is the unit disk u²+v²<1, which maps the circular arcs _k={w=eⁱ:_k<<_K+1} (k=1,..., N+3) into the straight lines containing the edges a_k, a_k+1 (a_N+4=a₁) of a polygon IR^p. Then we show that the function ()= _inf ^xF() D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

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Hans Lewy und Charles B. Morrey gewidmet