On Finite Pseudorandom Binary Sequences, VI,(On Sequences)

 Let K be a positive integer and α be a real number, and for let if the fractional part of is , and if it is . The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. In a previous paper the authors studied the special cases and , while here the case is considered. (Received 5 November 1997; in revised form 7 March 2000)     

On the Average Discrepancy of Successive Tuples of Pseudo-Random Numbers over Parts of the Period

 In the present paper we give an upper and a lower bound for the average value of the discrepancy of non-overlapping s-tuples of successive elements of a first order congruential pseudo-random-number generator (with prime modulus and maximal period). The estimates are – up to logarithmic factors – sharp also for short parts of the period. Received 30 January 1997; in revised form 2 May 1997     

The Automorphism Group of a Compact Generalized Polygon has Finite Dimension

 For every compact generalized polygon whose point space has finite dimension, we show that the automorphism group has finite dimension as well. This opens the field for the application of Lie theoretic methods. Received 12 November 1997; in revised form 10 March 1998     

Homomorphisms on Lipschitz Spaces

 Denote by the family of all real valued functions on a metric space which satisfy a Lipschitz condition on the compact (bounded) subsets of X. We prove that every homomorphism on is the evaluation at some point of X if and only if X is realcompact (every closed bounded subset of X is compact). (Received 4 November 1998; in revised form 31 May 1999)     

Isometric Shifts and Metric Spaces

 Let M be a complete metric space. If admits an isometric shift, then M is separable. (Received 2 February 2001; in revised form 9 April 2001)     

Universal Köthe Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note. Received 15 January 1997; in final form 9 June 1997     

On the stability of the incomplete LU-factorizations and characterizations of H-matrices

Summary. Meijerink and van der Vorst [8] have shown that the incomplete LU-factorizations are numerically stable for M-matrices. Varga, Saff and Mehrmann [16] gave some characterizations of the H-matrices by using the incomplete LU-factorizations of them. The purpose of this paper is to show that the incomplete LU-factorizations of an H-matrix are at least as stable as the complete LU-factorizations of its comparison matrix. We give also some new characterizations of the H-matrices in connection with their incomplete LU-factorizations. Received November 12, 1993 / Revised version received May 27, 1994     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

Invertibility Conditions and Bounds for Spectra of Matrix Integral Operators

 Matrix integral operators are considered. Bounds for the spectrum are established. In particular, they give the invertibility conditions and estimates for the spectral radius. (Received 16 December 1998; in revised form 30 May 1999)     

Involutorische Automorphismen ebener konvexer Mengen

 A bijection of the compact convex set with is called a reflection of K, if σ maps convex subsets of K into convex subsets. Conditions are stated unter which the existence of few reflections imply that is an ellipse. Received 4 March, 1998     

On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems

Summary. We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence. Received March 3, 1992/Revised version received July 2, 1993     

On Patterns in Sequences of Random Events

 This paper considers patterns of particular events in sequences of trials, some independent and others Markovian. Matrix recursions are found for the number of sequences of length n avoiding a specific pattern, and the associated probability of this event is evaluated. A Markov chain method for the study of such problems is outlined, and is illustrated in various cases. Finally, configurations of length 3 in Bernoulli trials are examined as an example. Received 12 January 1998 in revised form 26 June 1998     

On Hardy-Type Inequalities and Hankel Transforms

 For the real Hardy spaces , we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities with respect to Hankel transforms. Received 21 October 1997; in revised form 19 October 1998     

On hybrid iterative methods for nonsymmetric systems of linear equations

Summary. Hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired, and a second phase in which a polynomial iteration designed with respect to this information is used. Most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the Arnoldi process in Phase I. We will show the limitations of this approach and propose an alternative, also based on the Arnoldi process, which approximates the field of values of the coefficient matrix and of its inverse in the Krylov subspace. We also report on numerical experiments comparing the resulting new method with other hybrid algorithms. Received May 27, 1993 / Revised version received November 14, 1994     

Formal Power Series and Some Theorems of J. F. Rittin Arbitrary Characteristic

 We study composition of power series and polynomials over algebraically closed fields of arbitrary characteristic. The so-called Boettcher function of a power series is introduced and investigated. It is the principal aim of this paper to prove some results going back to J. F. Ritt in this general setting. In particular, we determine the pairs of permutable polynomials and characterize polynomials which satisfy a certain rational functional equation and polynomials which have a common iterate. Received 12 February 1998; in revised form 20 April 1998     

Compactification of Limit ?-Net Spaces

 Limit ?-net spaces are defined as convergence spaces whose convergence is expressed by using generalized nets, the so-called ?-nets (where ? is a construct). For limit ?-net spaces we study compactifications, especially those ones that are analogous to the Alexandrov and Čech-Stone compactifications known for topological spaces. (Received 24 February 2000)     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

 Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data. Received 3 February 1997; in revised form 9 June 1997