On the completeness of a system of functions on a segment

The question is considered of the completeness of the systems of functions {Xⁿ[1÷_n]}, where _n(x) are small, in the spaces C and L_p on the segment [0, a].Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 557–568, November, 1968.     

Bayesian estimation in the symmetric location problem

Summary Let X _i =+ _i for i=1, ..., n, where the _i's are i.i.d. F and F is symmetric about 0. F is assumed unknown or only partially known, and the problem is to estimate . Priors are put on the pair (F,). The priors on F are obtained from Doksum's neutral to the right priors, and include symmetrized Dirichlet priors. The marginal posterior distribution of given X ₁, ..., X _nis computed and its general properties studied. It is found that for certain classes of distributions of the _i's, the posterior distribution of is for all large n a point mass at the true value of . If the distribution of the _i's is not exactly symmetric, the Bayes estimates can behave very poorly.     

Remarks on ergodicity of stationary irreducible transient Markov chains

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 {itX_n(), 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.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

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 .     

A remark on the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variablesX _i ,i=1, 2, 3,... with finite expectationE|X ₁| can be stated as, for any >0, the number of integersn such that \varepsilon $$ " align="middle" border="0"> ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that \varepsilon $$ " align="middle" border="0"> andE N ₁ ² =.     

Testing for Unit Roots in a Nearly Nonstationary Spatial Autoregressive Process

The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model _st= _s-1,t+_s,t-1- _s-1,t-1+_st. Moreover, for the sequence _n = e ^c/n , _n = e ^d/n of local Pitman-type alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.     

Construction of ɛ-nets

LetS be a set ofn points in the plane and let be a real number, 0<<1. We give a deterministic algorithm, which in timeO(n ^–2 log(1/)+ ^–8) (resp.O(n ^–2 log(1/)+ ^–10) constructs an-netNS of sizeO((1/) (log(1/))²) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).We also prove that given a setL ofn lines in the plane, we can cut the plane intoO( ^–2) triangles in such a way that no triangle is intersected by more thann lines ofL. We give a deterministic algorithm for this with running timeO(n ^–2 log(1/)). This has numerous applications in various computational geometry problems.     

Finite activation energy reactions Part II: A reactive ridge

Summary The effect is determined of a large,O( ^–2),1, activation energy on the thermal stability of a reactant in the form of a two-dimensional ridge, undergoing a steady zero-order exothermic reaction. The reactive ridge decreases in depth at a rate ofO( ²) from a maximum ofO(1). The Biot number of the uniform lower surface of the reactant is taken to be zero and of the upper surface to beO( ^–2). The critical Frank-Kamenetskii parameter is determined toO( ²).     

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

Asymptotic expansions for bivariate von Mises functionals

Summary A Berry-Essen result and asymptotic expansions are derived for the distribution of bivariate von Mises functionals under moment and smoothness conditions.The results apply to the Cramér-von Mises ² — statistic as well as to the Central Limit Theorem in Hilbert space, yielding a convergence rate O(n _–1+) for every >0 on centered ellipsoids.Herrn Professor Dr. Leopold Schmetterer zu Ehren seines sechzigsten Geburtstages gewidmet     

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rate-balanced. Sharp bounds on in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallest in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,-balanced for0.06.     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Random symmetric polynomials

Let {Q⁽ⁿ⁾(x₁,...,x_n)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {X_n1,...,X_nn}, n k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Y_n=Q⁽ⁿ⁾(X_n1, ... X_nn), n k One obtains limit theorems for the sequence Y_n, under very general assumptions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 170–188, 1986.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A model spectral problem of the form -i)y+xy= y on the finite interval [-1,1] with the Dirichlet boundary conditions is considered. Here is the spectral parameter and is positive. The behavior of the spectrum of this problem as 0 is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Simultaneous Extensions of Topologies Through Traces of Neighbourhood Filters

Let X be a set, X _i X for i I and, for x X, a filter in X _i. The paper gives necessary and sufficient conditions for the existence of a topology on X compatible with , i.e. such that is the trace on X _i of the -neighbourhood filter of x. It is shown that, among these compatible topologies,there are always a coarsest one and a finest one. Some applications are also given.     

On a conjecture of Herzer concerning linear partitions

We prove: Let G be a finite group with a non-trivial partition (G) and a set (G) of subgroups such that any two components of (G) are contained in at least one W with W (G) and W G; then G is a p-group or a Frobenius group.From this we obtain: A finite group with a linear partition can only be an abelian p-group or a Frobenius group.Dedicated to Prof. H. Lenz on the occasion of his 70^th birthday