Validity of the multiple scale method for very long intervals

For a class of nonlinear oscillation problems containing a small parameter, it is known that a two-scale method using timest and t gives results valid to any desired order for time (1/). We ask when results can be obtained which are valid for (1/²) or for allt > 0. We show that there is an obstruction to introducing a third time scale ² t, and give an example in which this obstruction does not vanish, so that a third scale cannot be introduced, even though the solution exists for all time. The obstruction does vanish if the first order averaged equation vanishes, in which case the three-scale solution actually involves onlyt and ² t and is valid for time (1/²). The obstruction also vanishes if a certain contracting or dissipative condition is met, but in this case the two-scale solution is already valid for all time and the third scale is not needed. These results correspond to known results for the method of averaging, but are here proved for the multiple scale method without use of averaging.     

SU2ℍ als Kollineationsgruppe von sechzehndimensionalen lokalkompakten Translationsebenen

As a contribution to a classification of all sixteen-dimensional translation planes whose collineation group has dimension at least 38, this paper deals with the case that contains a subgroup (locally) isomorphic to SU₂Spin₅. Under various further assumptions, it is shown that such a plane satisfying dim 38 is necessarily isomorphic to the classical plane over the octonions.The complete classification will reveal that these further assumptions may in fact be omitted, except for the case that even contains a subgroup isomorphic to Spin₇. The latter planes have been explicitly determined in previous papers.

Meinem verehrten Lehrer Helmut Salzmann zum 65. Geburtstag     

Some stability properties for analytic operator functions

Let be a connected, finite-dimensional, complex analytic manifold; let T() be an analytic function defined on , whose values are J-biexpanding operators on a J-space H. Let (A) denote the range of A. The following assertions are proved: 1. The lineals and do not depend on . 2. For arbitrary we have Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 511–520, October, 1976.     

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 =.     

$$\omega $$ -Fibered Formations and Fitting Classes of Finite Groups

In this paper, we suggest a new functional approach to the study of group classes which enables us to describe all formations and Fitting classes of finite groups in the language of functions. The notions of -fibered formation and of -fibered Fitting class with direction are introduced. A direction is defined as a mapping of the set of all primes into the set of all nonempty Fitting formations. The existence of infinitely many mappings of this kind makes it possible to construct new forms of formations and Fitting classes for a given nonempty set . In particular, an -local formation is an -fibered formation with a direction such that for any prime p. In the paper we study some basic properties of -fibered formations and of -fibered Fitting classes with direction and obtain the structure of their minimal satellites for a given .     

The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array of i.i.d. random variables with mean and variance and set . Let denote the empirical distribution function of Z_{1, n} ,..., Z _{N, n} and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for , where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.     

Convergence of orthogonal series to K+∞

For any sequence {b_n} such that. =, a uniformly bounded system _n(x)} orthonormal on [0, 1], is constructed such that the series diverges to + on some set E [0, 1], 0<mes E<1, for any order of the terms.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 129–136, August, 1970.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

On the order of partial sums of general orthogonal series

It is shown that for convergence of every orthonormal system _n(x) given on [0, l],it is necessary and sufficient that, under the condition on tlie increasing function W(x) and for there hold almost everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 451–462, October, 1969.     

On Some Proximinal Subspaces of Modular Spaces

Let (, , ) be a complete measure space, L⁰ the vector lattice of -measurable real functions on , : L⁰ [0, )] a lattice semimodular, the corresponding modular space, S₀ the ideal generated by and 0,{\text{ }}\exists {\text{ }}s \in {\text{ }}S_{\text{0}} {\text{ such that }}\rho \left( {\frac{{x - s}}{\user1{\lambda }}} \right) < \infty } \right\}$$ " align="middle" border="0"> . In X consider the distance 0:\rho \left( {\frac{{x - y}}{\user1{\lambda }}} \right) \leqq \user1{\lambda }} \right\}$$ " align="middle" border="0"> and, if is convex, the distances d_L, d_o subordinated to the Luxemburg and Amemiya-Orlicz norms, respectively. We give necessary and sufficient conditions for H(S_o) in order to be proximinal in X with the distances d, d_L and d_o.     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

Approximation by Simplest Fractions

In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions on compact subsets of the complex plane by simplest fractions of the form are considered. In particular, it is shown that the best approximation of a function by the fractions is of the same order of vanishing as the best approximations by polynomials of degree .     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

A Criterion for Contiguity of Quasiconcave Functions

Quasiconcave functions and belong to the same scale if there exist quasiconcave functions and and numbers such that and . We establish a criterion for such functions to belong to the same scale up to equivalence. This criterion is obtained in terms of nodes of the corresponding linear-constant step-functions. It turns out that nodes must be equivalent to sequences.