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1.
We give an estimate for the quantity {f(n):nx, p(n)y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, ({(n):nx, p(n)y}) ({1:nx, p(n)y})–1 tends to zero, asx, uniformly iny2, and thus settle a conjecture of Erdös.Supported by a grant from the Deutsche Forschungsgesellschaft.  相似文献   

2.
Summary LetPQ ben×n real matrices so that ifPAQ for some matrixA, thenA is nonsingular. Letp andq ben-dimensional real column vectors. This paper determines the set of all solutionsx to the equationAx=b for allA andb so thatPAQ andpbq.  相似文献   

3.
A relation between Chung's and Strassen's laws of the iterated logarithm   总被引:2,自引:0,他引:2  
Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)–1/2f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)–1/2 W(xT)¦(loglog T)–1]=/4 a.s. T 0x1  相似文献   

4.
Summary Consider a stationary process {X n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X n(), - < n < and of each of the subprocesses {X n(), 0 n < , {X n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X n(), - < n <} with {itXn(), 0 n < < ergodic but {X n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X n(), 0 n < and {itX n(),-< < n 0} ergodic but {X n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.  相似文献   

5.
Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.  相似文献   

6.
Let p be an odd prime. For each integer a with x < a x + u and (a,p) = 1, we define by a 1 (mod p) and 1 p - 1. Let r(p,u,x) be the number of integers with x < a x + u and (a,p) = 1 for which a and are of opposite parity, and let E(n,u,x) = r(n,u,x) - 1/2 1, where denotes summation over all a such that (a,p) = 1. The main purpose of this paper is to prove that for any positive integer 1 u we have the asymptotic formula
.  相似文献   

7.
Summary There have been many studies of the values taken on by continued fractionsK(a n /1) when its elements are all in a prescribed setE. The set of all values taken on is the limit regionV(E). It has been conjectured that the values inV(E), are taken on with varying probabilities even when the elementsa n are uniformly distributed overE. In this article, we present the first concrete evidence that this is indeed so. We consider two types of element regions: (A)E is an interval on the real axis. Our best results are for intervals [–(1–), (1–)], 0 <1/2. (B)E is a disk in the complex plane defined byE={z:|z|(1–)}., 0<1/2.  相似文献   

8.
Summary LetC be the symmetric cusp {(x, y)2:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.  相似文献   

9.
Summary The optimal extreme-discrepancyN-point sets in the unit square are found forN6. The point ordering is unique forN3, but there are 14, 29 and 22 distinct orderings up to reflection about the diagonal forN=4, 5 and 6, respectively. The minimum discrepancy is less than, equal to, and greater thanN –1 forN3, 4N6 andN7, respectively.This work was performed at the University of Wisconsin with support from the National Science Foundation.  相似文献   

10.
Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt 0 and >0 such thatZ(t 0) is an extremal point of the convex hull of {Z(t)|t 0–tt0} and also an extremal point of the convex hull of {Z(t)|t 0tt0+} and, moreover, the tangent lines to the convex hulls atZ(t 0) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t 0 and >0 such that the intersection of the convex hulls of {Z(t)|t 0–tt0} and {Z(t)|t 0tt0+} contains onlyZ(t 0)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture  相似文献   

11.
One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions 1 and 2 inL () such that j ,j=1,2 is a short range perturbation of j when (–1) j x n +. One assumes j = (j) 1R,j=1,2, with (j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators j ± =s j e itHj ,j=1,2, withH j =–· j and j : equal to 1 if (–1) j x n >0 and to 0 if (–1) j x n <0. The ranges ofW j ± :=( j ± )* are characterized so that W 1 ± =Ran and . The scattering operator can then be defined.  相似文献   

12.
The Saturation Class of Shepard Operators   总被引:1,自引:0,他引:1  
The saturation problem of the Shepard operators for 1 2 is completely settled.  相似文献   

13.
Summary Let {W(t); 0t1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);rt1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.  相似文献   

14.
Summary This paper is concerned with the rate of convergence to zero of theL pmetrics np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, np/np is universally bounded away from zero and infinity asn.  相似文献   

15.
For a large real parameter t and 0 a b we consider sums where is the rounding error function, i.e. (z) = z - [z] - 1/2. We generalize Huxley's well known estimate by showing that holds uniformly in 0 a b . Fruther, we investigate an analogous question related to the divisor problem and show that the inequality , which (due to Huxley) holds uniformly in 0 a b , and which is in general not true for 1 a b t, is true uniformly in 0 a b .  相似文献   

16.
Summary The equation to be considered is of the form (1) x(n)(t)+p(t)x(g(t))=0 (t>a), where =±1, p(t) > 0 for ta and g(t) as t. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(i)(t)>0 (0il), (–1)i–lx(t)x(i)(t)>0 (lin) for some integer l, 0ln, (–1)n–l–1=1. In this paper, for a given l such that 0n–l–1=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every >0 the equation x(n)(t)+p(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.  相似文献   

17.
Given a function: + on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with f, where f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)–1 f, where M is the basis constant of E. In the particular case of E= 1 there are holomorphic functions f on with= f.  相似文献   

18.
Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.  相似文献   

19.
The problem of optimal choice of knots is considered for the functions belonging to the classW 2m+1 V, concerning interpolation by means of Hermite splines. The problem of asymptotically best choice of the knots for interpolation of a fixed functionf(x) (f(2m+2)(x)>0, 0x1) by Hermite splines is also treated.  相似文献   

20.
Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Muu and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea 2 and for given functions = we require =C 0[0, 1] C 2([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .  相似文献   

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