Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

Optimal Controller for the Almost Exact Guidance of a Singularly Perturbed System to Zero

The method of penalty functions is used to construct an optimal controller that guides a singularly perturbed system (vehicle) into a small neighborhood of a specified (target) position. It is proved that the convergence of the path of the vehicle to the exact-guidance path as the penalty coefficient increases without bound is uniform with respect to a singular parameter.     

On the exactness of a theorem of G.A. Fomin

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Асимптоический Анализ Распределения Статистики Диксона

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Morera type theorems on the unit disc

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Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

О числе Делителей “Сдвинутых” Простых чисел-Близнецов

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A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

On the parameter estimation of diffusional type processes with constant coefficients (Elementary Gaussian Processes)

, (1). 3, , ()=, (8) (16). [1], . (28) (31) ( 5), - (. [3]).

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The author thanks Professor M.Arató for having pointed out this problem, and for his valuable suggestions.     

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Let be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under consists of compact operators. The same is true if one replaces compact by nuclear, Hilbert-Schmidt or positive Fredholm. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

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Gefördert von der Deutschen Forschungsgemeinschaft im Rahmen des Forschungsvorhabens Ko 506/1.     

Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

О Сходимости Тригонометричецкого и Гармонического интерполирования

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Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Duality for nonconvex optimization and its applications

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About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

Direct and converse imbedding theorems for seminormed spaces

. n=2 . ., 103 498      

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.