Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen

In [8] the author extended the concept of neighbouring functions (cp. [9]) to the case of several variables. Using these results it is shown that under some weak conditions a multiplicative functionf in two variables has a mean-value different from zero if and only if the two multiplicative functionsf ₁(n)=f(n, 1) andf ₂(n)=f(1,n) have mean-values different from zero. Applications to theorems ofDelange [3],Elliott [6] andDaboussi [1] are given.     

Fano'sche Moufang-Ebenen mit Gauss-Figur sind pappos'sch

LetV be a quadrilateral in aMoufang-plane , in which theFano-proposition is valid. Take the pointsP,Q,R respectively in the diagonalsp,q,r ofV and construe the pointsP ^*,Q ^*,R ^* inp,q, r harmonic toP,Q,R with respect to pairs of edges ofV. IfP,Q,R are collinear, so areP ^*,Q ^*,R ^*, if and only if is aPappos-plane. Is V classical, the pointsP ₁ p,Qq,Rr and their harmonic conjugatesP ₁ ^* ,Q ^*,R ^* (construed as above mentioned) lay in a curve of 2^nd order.

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R. Artzy zum 70. Geburtstag zugeeignet     

Maximum Monoreflections and Essential Extensions

An extension operator c in a category is an assignment, to each object A a monomorphism c _A : AcA. Seeking to approximate such a c by a functor, in our earlier paper Maximum monoreflections, we showed that with some hypotheses on the category, and on c, there is a monoreflection (c) maximum beneath c. Thus, in a suitable category of rings, using the complete ring of quotients operator Q, each object A has a maximum functorial ring of quotients (Q)A. But the proof gave no hint of how to calculate the general (c)A's, nor the particular (Q)A's. In the present paper, we give an explicit formula (and separate proof of existence) for the (c)A's, under more complicated hypotheses on the category and assuming the c _A 's are essential monomorphisms. We discuss briefly how the formula proves adequate to calculate the (Q)A's in Archimedean f-rings, and some related and necessary constructs in Archimedean l-groups.     

The algebraic radical of a normed ideal inL 1(G)

LetA be a proper normed ideal (in the sense ofCigler) insideL ¹ (G), whereG is a non-discrete LCA group. This is proved: For each integern1 there existsfL ¹ (G) such thatf, f ² ,..., f ^nA whilef ⁿ⁺¹ A.     

On Locating Clusters of Zeros of Analytic Functions

Given an analytic function f and a Jordan curve that does not pass through any zero of f, we consider the problem of computing all the zeros of f that lie inside , together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along . If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational interpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques.     

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.     

The periodic Dirac operator is absolutely continuous

We consider the periodic Dirac operatorD inL ₂( ^d ). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,VL _{r, loc} ,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrödinger operatorM. Ford3 our considerations are based on the estimates forM, obtained in [So] forAC ^2d+3 . Under the same condition onA, forVC, the absolute continuity ofD, d3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].The research was completed in the framework of the project INTAS-93-351.     

Convergence of theQR algorithm

TheQR algorithm ofJ. G. F. Francis is used in computing matrix eigenvalues. The convergence proof given here is an analogue ofRutishauser's proof of the convergence of theLR algorithm, but our proof covers the case of disorder of the eigenvalues.The work presented in this paper was supported by the AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, under contract AT(30-1)-1480 with the U.S.Atomic Energy Commission.     

Numerical studies of the Möbius power series

We define the Möbius power series throughf(z)= _n-1 z ⁿ ,g(z)= _n=1 (n)z ⁿ /n where (n) is the usual Möbius function. This paper presents some heuristic estimates describing the behavior off(z) andg(z) when |z| is close to 1 together with representations in terms of elementary functions for real values ofz. Function tables are also given together with zeros and a few other special values.     

How to cage an egg

Summary This paper proves that given a convex polyhedronP ³ and a smooth strictly convex bodyK ³, there is some convex polyhedronQ combinatorically equivalent toP whichmidscribes K; that is, all the edges ofQ are tangent toK. Furthermore, with some stronger smoothness conditions on K, the space of all suchQ is a six dimensional differentiable manifold.Oblatum 18-V-1991     

Inequalities for the interpolation points in Chebyshev approximation by polynomials

Letf andg be approximated in the Chebyshev sense by polynomials of degree n and n–1, respectively. It is shown that if the sum and difference of the normalized (n+1)-st derivatives off andg do not change sign, then the interpolation points ofg separate those off. A corollary is that the zeros of the Chebyshev polynomialT _n separate the interpolation points off iff ⁽ⁿ⁺¹⁾ does not change sign. The sharpness of this result is demonstrated.     

Numerical study of the representation of a totally positive quadratic integer as the sum of quadratic integral squares

Resume Altogether a total of almost 200,000 totally positive numbers from different fields were decomposed into squares; in no case were more than five squares required, although in many cases no number of squares sufficed. For some quadratic fields, from our evidence it would seem safe to completely characterize the couples for whichQ=5 or 0, particularly whenm13; and furthermore, form=17 or 33 it seems possible to characterize all cases whereQ=4, 5, or 0.The whole calculation seems to be pointed toward the result that three squares are sufficient except for special cases. Incidentally, the analytic methods ofSiegel andMaass run parallel to the calculation in that these methods involve the third, fourth, and fifth power of a theta-function. The numberical evidence would therefore suggest that their methods point to an analytic (or even a purely algebraic) proof of the futility of using more than five squares in any case.The work was supported in part by the U. S. National Science Foundation Grant G-4222 and the computer services were contributed by the Argonne National Laboratory of the U. S. Atomic Energy Commission during the summer of 1958. The coding was performed by Mr.Alan V. Lemmon with remarkable economy of length of program and running time.The deepest debt of gratitude is owed to the lateDonald A. (Moll) Flanders whose contributions to the logical design ofGeorge had made the rapid execution of the program possible and whose personal interest made possible the availability of the computer for this work. Donald A. Flanders in Memoriam (1900–1958).     

Minty Variational Inequalities,Increase-Along-Rays Property and Optimization<Superscript>1</Superscript>

Let E be a linear space, let K E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^,K ), which can be considered as a generalization of MVI (f ^,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^K ) and the property of f to increase-along-rays starting at x ^*, fIAR (K,x ^*). We prove that the GMVI (f ^,K ) with radially l.s.c. function f has a solution x ^* ker K if and only if fIAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^,K ) has a solution x ^*K, then x ^* is a global minimizer of the problem min f(x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.     

A compactness property of Dedekind σ-completef-rings

We prove that Dedekind -completef-rings are boundedly countably atomic compact in the language (+, –, ·,, , ). This means that whenever is a countable set of atomic formulae with parameters from some Dedekind -completef-ringA every finite subsystem of which admits a solution in some fixed productK of bounded closed intervals ofA, then admits a solution inK.Presented by M. Henriksen.     

On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Recently, Mandelbrot has encountered and numerically investigated a probability densityp _d (t) on the nonnegative reals, where, 0D<1. this=" density=" has=" fourier=" transform=">f _d (-is), wheref _d (z)=–Dz ^d (–D, z) and (·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressf _d (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatep _d (t) asymptotically ast0+ and +, then note some implications asD0+ and 1–.Communicated by Mourad Ismail.     

Fehlerabschätzungen für nichtsymmetrische Gauß-Quadraturformeln

Summary We consider Gaussian quadrature formulaeQ _n , n, approximating the integral , wherew is a weight function. In certain spaces of analytic functions the error functionalR _n :=I–Q _n is continuous. Previously one of the authors deduced estimates for R _n for symmetric Gaussian quadrature formulae. In this paper we extend these results to nonsymmetric Gaussian formulae using a recent result of Gautschi concerning the sign ofR _n (q _K ),q _K (x):=x ^K , for a wide class of weight functions including the Jacobi weights.

     

Degree of approximation of analytic functions by “near-best” polynomial approximants

We study the rate of approximation of a functionfA(K) in the interior of the compact setK by polynomials, that are close to best polynomial approximants on the whole setK. Lower and upper estimates of possible improvement of convergence (depending on the geometry ofK) are obtained.Communicated by Vilmos Totik.     

Approximation and regular perturbation of optimal control problems via Hamilton-Jacobi theory

We present two convergence theorems for Hamilton-Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. LetT andT _h be the minimal time functions to reach the origin of two control systemsy = f(y, a) andy = f _h (y, a), both locally controllable in the origin, and letK be any compact set of points controllable to the origin. If f _h –f Ch, then |T(x) – T _h (x)| C _K h , for all x K, where is the exponent of Hölder continuity ofT(x).     

Inverspositive Operatoren und Taubersätze in der Erneuerungstheorie

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ₊ the aim is (a,a+) with 0a<, 0< instead of (a, ), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a0(a), with fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchröder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBene.

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Herrn Professor Collatz zum 65. Geburtstag     

Root determination by use of Padé approximants

In this paper we describe a new technique for generating iteration formulas — of arbitrary order — for determining a zero (assumed simple) of a functionf, assumed analytic in a region containing the zero. The 1/p Padé Approximant (p0) to the functiong(t)f(z) is formed wherez=w+t, using the Taylor series forf at the pointw, an approxination to the zero off. The value oft for which the 1/p Padé Approximant vanishes provides the basis of iteration formulas of orderp+2.Some known iteration formulas, e.g., Newton-Raphson's, Halley's and Kiss's of order of convergence two, three and four, are directly obtained by settingp=0,1 and 2, respectively.