On σ-discrete borel mappings via quasi-metrics

Let X and Y be metrizable spaces. We show that, for a mapping f : X Y, there exists a quasi-metric X inducing the topology of X such that f regarded as a mapping from (X, max{, ^–1}) to Y is continuous if and only if f in the original topology of X is a -discrete map of Borel class 1. Further, we prove that, for every -discrete mapping f: X Y of Borel class + 1, there exists a compatible quasi-metric on X such that f : (X, max{, ^–1}) Y is of Borel class . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Algorithms for optimization problems with exclusion constraints

This paper proposes algorithms for minimizing a continuously differentiable functionf(x): ⁿ subject to the constraint thatx does not lie in specified bounded subsets of ⁿ . Such problems arise in a variety of applications, such as tolerance design of electronic circuits and obstacle avoidance in the selection of trajectories for robot arms. Such constraints have the form . The function is not continuously differentiable. Algorithms based on the use of generalized gradients have considerable disadvantages because of the local concavity of at points where the set {j|g ^j (x)=(x)} has more than one element. Algorithms which avoid these disadvantrages are presented, and their convergence is established.This research was sponsored in part by the National Science Foundation under Grant ECS-81-21149, the Air Force Office of Scientific Research (AFSC), United States Air Force under Contract F49620-79-C-0178, the Office of Naval Research under Grant N00014-83-K-0602, the Air Force Office of Scientific Research under Grant AFOSR-83-0361, and the Semiconductor Research Consortium under Grant SRC-82-11-008.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

We investigate the multidimensional equations _j=1 ^q A_j(x)y(x+e _j )=f(x),e _j ⁿ wherex ⁿ andA _j : ⁿ Hom( ^p , ^m ),f : ^{n m} are given maps. Sufficient conditions for smooth and analytic solvability for anyf C ^k ,k are found.Research partially supported by the Israel Ministry of ScienceAMS classification 39B Functional equations     

On dominant operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T–)f() x (T a dominant or hyponormal operator on the Hilbert space ,x andf a function from the open setU to ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.Supported in part by the National Science Foundation.     

Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization

We propose an algorithm for minimizing a functionf on ⁿ in the presence ofm equality constraintsc that locally is a reduced secant method. The local method is globalized using a nondifferentiable augmented Lagrangian whose decrease is obtained by both a longitudinal search that decreases mainlyf and a transversal search that decreases mainly c. Our main objective is to show that the longitudinal path can be designed to maintain the positive definiteness of the reduced matrices by means of the positivity of _k ^T _k , where _k is the change in the reduced gradient and _k is the reduced longitudinal displacement.Work supported by the FNRS (Fonds National de la Recherche Scientifique) of Belgium.     

Continuity of martingales in the Brownian excursion filtration

Summary Given a Brownian motionB, we consider the filtration ( _{x xR} ), where _x is defined as the -field generated by the excursions ofB belowx. In this paper we prove a conjecture of Walsh which says that all -martingales are continuous.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

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.     

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.     

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.     

Sequential gradient-restoration algorithm for the minimization of constrained functions—Ordinary and conjugate gradient versions

The problem of minimizing a functionf(x) subject to the constraint (x)=0 is considered. Here,f is a scalar,x ann-vector, and aq-vector. Asequential algorithm is presented, composed of the alternate succession of gradient phases and restoration phases.In thegradient phase, a nominal pointx satisfying the constraint is assumed; a displacement x leading from pointx to a varied pointy is determined such that the value of the function is reduced. The determination of the displacement x incorporates information at only pointx for theordinary gradient version of the method (Part 1) and information at both pointsx and for theconjugate gradient version of the method (Part 2).In therestoration phase, a nominal pointy not satisfying the constraint is assumed; a displacement y leading from pointy to a varied point is determined such that the constraint is restored to a prescribed degree of accuracy. The restoration is done by requiring the least-square change of the coordinates.If the stepsize of the gradient phase is ofO(), then x=O() and y=O(²). For sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionf decreases between any two successive restoration phases.This research, supported by the NASA Manned Spacecraft Center, Grant No. NGR-44-006-089, and by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensation of the investigations reported in Refs. 1 and 2.     

Algorithm for a sum of reciprocals

An algorithm is described for the approximate calculation of a collection of sums of the form _k= _j–1 ⁿ c_j/(_j+_k), 1kn, where 0<_j. The working time of the algorithm is 0(n(t+ log n)(t+log n)) if _k calculated to within 2^–t; here the function (l) denotes the time of multiplication of twoZ-bit numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 3–6, 1984.In conclusion, the author expresses thanks to A. O. Slisenko and Yu. A. Kuznetsov, who drew the attention of the author to the problem considered.     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

On dense sequences of polynomials in several variables

In the present paper a form of generalization of Gelfond's lemma on dense sequences of polynomials is proposed. For a set of complex numbers ₁, ..., _s we define the coefficients_gk( ₁, ..., _s ) (0ks) and give the relations between them and the transcendental degrees or the transcendence types of the field © ( ₁, ..., _s ) or its subfields.This work was completed at the Dept. of Math., Univ. of Southern Mississippi, USA in 1987.     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.