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1.
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.  相似文献   

2.
Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.  相似文献   

3.
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 .  相似文献   

4.
Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] b a [1 + (p(x)) 2]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.  相似文献   

5.
Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.  相似文献   

6.
Summary A nonlinear generalizationÊ z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE z and a twoparametric methodE . It is shown how to applyE orE , to compute the valuef(zo) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE , are superior toÊ z . A compact recursive algorithm is given for computingE andE ,.  相似文献   

7.
Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H2 H2 with the space L2 ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.  相似文献   

8.
Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH inf0 sup1 of Au=(f i ) x i , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).  相似文献   

9.
Summary Considerf+ ff+ (1–f2)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.
Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f2)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).
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10.
Harold L. Putt 《Order》1984,1(2):173-185
In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].  相似文献   

11.
For a mean zero norm one sequence (f n )L 2[0, 1], the sequence (f n {nx+y}) is an orthonormal sequence inL 2([0, 1]2); so if , then converges for a.e. (x, y)[0, 1]2 and has a maximal function inL 2([0, 1]2). But for a mean zerofL 2[0, 1], it is harder to give necessary and sufficient conditions for theL 2-norm convergence or a.e. convergence of . Ifc n 0 and , then this series will not converge inL 2-norm on a denseG subset of the mean zero functions inL 2[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c n | = 0(n ) for >1/2, then converges a.e. and unconditionally inL 2[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL p[0, 1] with 1p<. Finally, if (f n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x n )[0, 1], converges for a.e.y and in allL p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.  相似文献   

12.
Arató  N.  Márkus  L. 《Analysis Mathematica》1986,12(4):307-312
Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.  相似文献   

13.
If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · E(M,) is not proportional to · L 2, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council  相似文献   

14.
For a bounded regular Jordan domain in R 2, we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ,>0 with nonnegative functions qK(). Some estimates on the solution are also given.  相似文献   

15.
An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = 1 (x),..., r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector 0, exact estimates are obtained of the norms f(x)-s (f, 0; x)C[0,1] and f (x)-s (f, 0; x)L[0, 1] on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.  相似文献   

16.
Summary Given a complex polynomialp we determine a functionf p : such that |p(f p (z))||p(z)|,z withk<1. This result is used to introduce a global root-finding algorithm for polynomials.  相似文献   

17.
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.  相似文献   

18.
The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.  相似文献   

19.
Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV 0 in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA M (V 0)={xV 0 f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV 0 are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.  相似文献   

20.
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