Approximation of Continuous Functions by de La Vallee-Poussin Operators

We study the asymptotic (as σ → ∞) behavior of upper bounds of the deviations of functions belonging to the classes and from the so-called de la Vallee-Poussin operators. We obtain asymptotic equalities that, in some important cases, give a solution of the Kolmogorov-Nikol’skii problem for the de la Vallee-Poussin operators on the classes and .__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 230–238, February, 2005.     

A standard form variant,and safeguarded linesearch,for the modified Karmarkar algorithm

In his original analysis of the projective algorithm for linear programming, Karmarkar proposed a modified method which improved the worst-case arithmetic complexity of the original algorithm by a factor of . Karmarkar's analysis of the improvement is based on a primal-dual formulation, and requires that small steps be taken on each iteration. However, in practice the original algorithm can be easily applied to standard form problems, and is considerably improved by performing a linesearch on each iteration, generally leading to much larger steps. We show here that by incorporating a simple safeguard, a linesearch may be performed in the modified algorithm while retaining the complexity improvement over the original algorithm. We then show that the modified algorithm, with safeguarded linesearch, can be applied directly to a standard form linear program with unknown optimal objective value. The resulting algorithm enjoys a complexity advantage over standard form variants of the original algorithm.     

Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

We introduce the notion of (, )-derivative of a function of one complex variable, and define on the basis of this the classes of (, )-differentiable analytic functions in a bounded domain G. The classes consist of the Cauchy-type integrals whose densities f() are such that the induced functions on the unit circle are periodic functions of classes . We consider approximation of functions by algebraic polynomials constructed from their series expansions in Faber polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1556–1570, November, 1992.     

Approximation of the Classes $$C^{{\bar \psi }} H_{\omega }$$ by de la Vallée-Poussin Sums

We investigate the problem of the approximation of the classes introduced by Stepanets in 1996 by the de la Valée-Poussin sums. We obtain asymptotic equalities that give a solution of the Kolmogorov–Nikol'skii problem for the de la Valée-Poussin sums on the classes in several important cases.     

Extensions of hereditary algebras

Let be a triangular matrix algebra, uhere k is an algebraically closed field, B is the path algebra of an oriented Dynkin diagram of type E₆ or E₇ or E₈ and M is a finite dimensional k-B-bimodule. The aim of this paper is to determine the representation type of A for any orientation of the Dynkin diagram and for any indecomposable B-module M. This classification is obtained by comparing the representation types of the algebras and using the theory of tilting modules.     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

Prym Representations of Mapping Class Groups

Let S be a closed orientable surface of genus at least 2 and let to S be a connected finite abelian covering with covering group $G$. The lifts of liftable mapping classes of S determine a central extension (by G) of a subgroup of finite index of the mapping class group of S. This extension acts on H₁( ). With a few exceptions for genus 2, we determine the Zariski closure of the image of this representation, and prove that the image is an arithmetic group.     

Construction of Bases in Spaces of Solenoidal Vector Fields

We describe a method of construction of fundamental systems in the subspace H(Ω) of solenoidal vector fields of the space (Ω) from an arbitrary fundamental system in. ₂ ¹ (Ω). Bibliography: 9 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 92–106.     

Stability of invariant linearly sufficient statistics in the general Gauss-Markov model

Necessary and sufficient conditions are derived for the inclusions and to be fulfilled where are some classes of invariant linearly sufficient statistics (Oktaba, Kornacki, Wawrzosek (1988)) corresponding to the Gauss-Markov models , respectively.     

Young Straightening in a Quotient Sn-Module

We describe a straightening algorithm for the action of S _n on a certain graded ring . The ring appears in the work of C. de Concini and C. Procesi [2] and T. Tanisaki [8], and more recently in the work of A. Garsia and C. Procesi [4]. This ring is a graded version of the permutation representation resulting from the action of S _n on the left cosets of a Young subgroup. As a corollary of our straightening algorithm we obtain a combinatorial proof of the fact that the top degree component of affords the irreducible representation of S _n indexed by .     

Self-dual bases in\mathbb{F}_{q^n }

We characterize weakly self-dual bases of the field extension over , examine the existence of weakly self-dual polynomial bases, and use duality to analyze normal basis multiplication.     

The lift on a small sphere touching a plane in the presence of a simple shear flow

The contribution from purely viscous forces to the liftL on a sphere of radiusa touching a plane in the presence of a shear flow field of strength is zero. An exact integral expression for the lift to leading order in the Reynolds number is derived using known creeping flow solutions to related problems. The integral is evaluated numerically to obtain the value of the lift orL/F _x 0.287R whereF _x is the lateral viscous force on the sphere.

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Zusammenfassung Der Beitrag der Zähigkeit allein zum AuftriebL an einer Kugel vom Radiusa, welche in einer Scherströmung mit dem Gradienten eine unendliche Ebene berührt ist null. Ein exakter Integralausdruck für den Auftrieb wird in erster Ordnung der Reynoldszahl, , hergeleitet unter Benützung bekannter verwandter Lösungen in schleichender Strömung. Der Wert des Integrals wird numerisch bestimmt und gibt für den Auftrieb oderL/F _x 0.287R; dabei istF _x die seitliche Zähigkeitskraft auf die Kugel.

     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

Stability Zones in Whitney's Extension Problem for Ultradifferentiable Functions

We establish a connection between the growth rate of weight functions generating nonquasianalytic classes of ultradifferentiable functions of Beurling and Roumieu type and the validity of an analog of Whitney's extension theorem for these classes.     

Some estimates for the oscillation of the deformation gradient

As a measure of deformation we can take the difference D - R, where D is the deformation gradient of the mapping and R is the deformation gradient of the mapping , which represents some proper rigid motion. In this article, the norm is estimated by means of the scalar measure e( ) of nonlinear strain. First, the estimates are given for a deformation W ^1,p() satisfying the condition . Then we deduce the estimate in the case that (x) is a bi-Lipschitzian deformation and .     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. II

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic -, -, and -connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

Integration with respect to local time

Let be the local time process of a linear Brownian motion B. We integrate the Borel functions on with respect to . This allows us to write Itôrs formula for new classes of functions, and to define a local time process of B on any borelian curve. Some results are extended from deterministic to random functions.     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

On 2-Homogeneity of Monounary Algebras

Fraïssé introduced the notion of a k-set-homogeneous relational structure. In the present paper the following classes of monounary algebras are described: --the class of all algebras which are 2-set-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively, and --the class of all algebras which are 2-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively.