Bahadur asymptotic efficiency of ω2 -type tests in the many-sample case

Consider independent samples of sizen ₁,n ₂,... ,n ₂ and the empirical distribtion constructed using these samples. We test the hypothesis that all the samples are drawn from a population with the same continuous distribution function F (x). The test statistic is the weight function) is defined as generalizes Kiefer's well-known statistic originally proposed for the case K=2 and q,= 1. A rough asymptotics is obtained for the probability of large deviations of the statistic , which allows to give explicit expressions for the Bahadur local exact slopes and to compare the statistics for various K and q in the sense of Bahadur efficiency. In conclusion, the question posed by Renyi is considered: to what extent is it advisable to use statistics of the type instead of pooling all the samples and using a one-sample test?     

On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .     

Metrische Räume mit Dreiseitverbindbaren

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group O _n+1 ^* (K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups o _n+1 ^* (K,f) are models of the system of axioms.     

For which p and is the equation true?

It is explained when the classes of p-absolutely summing and p-integral operators given on the space coincide. For a Banach space there is considered the following subset of the real line: . In the case when is an infinite-dimensional subspace of the space , it is proved that J_X=(1,2] if , and J_X?{2} if and χ is not isomorphic with a Hilbert space.     

Completely- matrices

A matrix is a real square matrixM such that for everyq the linear complementarity problem: Findw andz satisfyingw = q + Mz, w ≥ 0, z ≥ 0, w ^T z = 0, has a solution. We characterize the class of completely- matrices, defined here as the class of -matrices all of whose nonempty principal submatrices are also -matrices.     

Best one-sided approximation of some classes of smooth functions on the whole real axis by cardinal splines of higher order

The objective of this paper is to study the quantity . In this paper we establish a duality theorem for the quantity for m≥r. By virtue of the duality theroem we obtain the exact estimation for for all m≥r.     

A class of limit distributions for maximum cumulative sum

Let X₁, X₂, ... be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variances. S_o=0 and S_n=∑ _i=1 ⁿ X_i. It is proved that is the limit distribution function of the normalized random variable a(k, n)} for some sequence of centering constants a (k,n).     

Extension of functions from Sobolev spaces

By definition, the domain Ω ??ⁿ belongs to the class EW _p ^l if there exists a continuous linear extension operator . An example is given of a domain Ω ??² with compact closure and Jordan boundary, having the following properties: (1) The curve ?Ω is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) Ω ε EW _p ¹ for p<2. and Ω ? EW _p ¹ for p?2. (3) for p>2 and for p?2.     

Der spektrale Abbildungssatz für nichtanalytische Funktionalkalküle in mehreren Veränderlichen

In this note we continue the study of -functional calculi in several variables (introduced in [1] and [2]). In the situation of an inverse closed (see Def. 1.2) basic algebra of type I or II we characterize the algebra generated by an arbitrary -functional calculus φ and prove several variants of the spectral mapping theorem, thus generalizing corresponding results of F.-H. Vasilescu [10] (see also the more general version in [4], Th. 3.2.1) to the case of several commuting operators. By means of the spectral mapping theorem we prove the decomposability (in the sense of St. Frunz? [6]) of m-tuples of the type (φ(f₁),..., φ(f_m)) with f₁,...,f_mε .     

Analytic continuation from a continuum to its neighborhood

Let be a positive number. A function f analytic in an open set ? ?? is called -analytic on the set E, E ??, if . THEOREM. Let K be a compact connected subset of the plane. For any >0 there exists an open neighborhood V of the set K such that any function -analytic on K coincides in some neighborhood of the set K with a function analytic in V. This theorem answers a question posed in the collection (RZhMat., 1979, 3B536, pp. 33–35 of the book).     

The rational approximation of convex functions of the class Lip α

It is proved that if a function f(x) is convex on [a, b] and f ∈ Lip_K(f)α, 0<α<1, then the least uniform deviation of this function from rational functions of degree not higher than n does not exceed (v is a natural number; C(α, v) depends only onα andv; K(f) is a Lipschitz constant; and      

On a lemma of Butzer and Kirschfink

In [1], Butzer and Kirschfink discussed the convergence rates for C[0,1]-valued dependent random functions on Donsker's weak invariance principle and introduced the concept of dependency from below to deal with the martingale difference sequence. They asserted in their Lemma 8 that Lemma A. A martingale difference sequence (X_n,F_n,n≥1) with is dependent from below, i.e., for each 1≤i≤n and each n≥1 . The purpose of this note is to prove that Lemma A is not always true and to improve the conditions of Butzer and Kirschfink. We shall apply the notations in [1].     

Ergodic properties of integral points on ellipsoids of genus

The ergodic and the intermixing theorems of Yu. V. Linnik (Mat. Sb.,43, No. 2, 257–276) are generalized to arbitrary positive quadratic forms of genus of , where Ω > 1 is an odd number, of invariants [Ω,1], defined by the character for all prime numbers ρ¦Ω. One obtains estimates for the remainder term. The method of proof is simplified.     

Interpolation by analytic functions smooth along the boundary. II

In this paper one considers the following classes of functions, analytic in the circle of the complex plane: , . One establishes that the well-known Carleson condition is necessary and sufficient for the solvability of the interpolation problems, intrinsic for the given class, on sets lying along a ray. In this connection, the class of sequences to be interpolated is described in terms of divided differences of order S. See also author's paper, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,30, 167–169 (1972).     

Modulus of boundary values of analytic functions of class

Let ω be a modulus of continuity, be the class of all functions analytic in the unit disk of the complex plane and such that . A condition is given (depending essentially on ω), necessary for a nonnegative function defined on the unit circle to coincide with the modulus of some function of class .