Zeros of orthogonal Laurent polynomials and solutions of strong Stieltjes moment problems

The strong Stieltjes moment problem for a bisequence consists of finding positive measures μ with support in [0,∞) such that

     

Equipartition of interval partitions and an application to number theory

We give wider application and simpler proofs of results describing the rate at which the digits of one number-theoretic expansion determine those of another. The proofs are based on general measure-theoretic covering arguments and not on the dynamics of specific maps.

     

Structure-dependent magnetic susceptibility of sharp poly(ethylene oxide) fractions

Nine sharp fractions of poly(ethylene oxide) (PEO) glycol with number-average molar masses (M _n) in the range from 0.6 × 10³ to 20 × 10³ (PEO-0.6 to PEO-20) were characterized by magnetic susceptibility χ measured in the temperature interval 293 K to 378 K. In contrast to the liquidlike PEO-0.6 with temperature-invariant χ, the values of χ for each of the remaining solid samples, after the initial increase, exhibited two plateaus separated by a relatively narrow temperature interval of their second increase. The jumps of χ at lower and higher temperatures were attributed to a solid-state transition of unspecific nature and to the melting of the crystal fraction, respectively.

The temperature-invariant values of χ_n in the melt state above T _m pass through a minimum for the sample PEO-2.0 and then increase again with (M_n) to a limiting value χ_∞ = ?0.622 × 10^?6. It is concluded that a considerable contribution of the molar-mass-dependent “paramagnetism” χ_P = χ ^? χ_d (where χ_d is the diamagnetic contribution estimated by Kirkwood's equation) to the total magnetic susceptibility of PEO fractions reflects distortions of the spherical symmetry of the electron shells around chain atoms resulting from the discontinuous change of both inter- and intrachain interactions as the (M_n) increases through and above the critical crossover molar mass (M_cr ) = 2 × 10³.     

The Hillam–Thron Theorem in Higher Dimensions

Let D be a disc in the complex plane, and let s _n be a sequence of Möbius maps each of which maps D into itself. In 1965 Hillam and Thron proved (essentially) that if the points s _n (), n=12,..., lie in a compact subset of D then the functions s ₁^...^s _n converge locally uniformly to a constant on D. They then made applications to continued fractions. In this paper, the corresponding results are proved in all dimensions.     

Arithmetics of binary quadratic forms,symmetry of their continued fractions and geometry of their de Sitter world

This article concerns the arithmetics of binary quadratic forms with integer coefficients, the De Sitters world and the continued fractions.Given a binary quadratic forms with integer coefficients, the set of values attaint at integer points is always a multiplicative tri-group. Sometimes it is a semigroup (in such case the form is said to be perfect). The diagonal forms are specially studied providing sufficient conditions for their perfectness. This led to consider hyperbolic reflection groups and to find that the continued fraction of the square root of a rational number is palindromic.The relation of these arithmetics with the geometry of the modular group action on the Lobachevski plane (for elliptic forms) and on the relativistic De Sitters world (for the hyperbolic forms) is discussed. Finally, several estimates of the growth rate of the number of equivalence classes versus the discriminant of the form are given.Partially supported by RFBR, grant 02-01-00655.     

The Vector Epsilon Algorithm – a Residual Approach

We present an alternative to the vector -algorithm based on vector continued fractions and which is applicable when the sequence to be accelerated is generated by a one-point iteration function. These fractions are constructed in the language of Clifford algebras, which allow three-term recurrence relations. The new algorithm evidently has considerably greater numerical precision than the old one. Results from numerical experiments are reported.     

The Toda molecule equation and the -algorithm

One of the well-known convergence acceleration methods, the -algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.

     

Convergence of random walks on the circle generated by an irrational rotation

Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of which allows for tighter bounds on terms which appear in the Erdös-Turán inequality.

     

On the Rogers-Ramanujan continued fraction

In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula.     

Variable-temperature study of a gel-spun ultra-high molecular-mass polyethylene fiber by solid state NMR

Solid-state¹³C nuclear magnetic resonance (¹³C NMR) methods have been applied to study a gel-spun ultra-high molecular-mass polyethylene fiber. The mass fractions of the crystalline (orthorhombic and monoclinic), intermediate, and amorphous phases have been determined at temperatures between 296 and 413 K. The mobility of the polymer chains in the crystalline and the intermediate phases have been measured within the same temperature range, 296–413 K. Discussions on the chain conformation and the rate of motion are carried out based on the¹³C chemical shift and spin-lattice relaxation time, respectively.Dedicated to Professor Bernhard Wunderlich on the occasion of his 65th birthday