Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

A note on idempotents in finite AW*-factors

We prove that the value of the quasi-trace on an idempotent element in an AW*-factor of type is the same as the dimension of its left (or right) support.

     

Centered -discrepancy of random sampling and Latin hypercube design, and construction of uniform designs

In this paper properties and construction of designs under a centered version of the -discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

     

Transient Nutation EPR Spectroscopy of Condensed Media (Review)

The application of transient nutations in EPR spectroscopy of condensed media is considered. The main methods of formation and observation of transient nutations are presented. The laws governing this phenomenon in twolevel and multilevel spin systems and also in inhomogeneous broadening of EPR lines are described. Recent advances in the use of transient nutations to separate overlapping spectra, identify quantum numbers and quantum transitions, investigate the kinetics of photoinduced paramagnetic centers, and determine relaxation times for a wide range of crystalline and disordered media are presented.     

The Asymmetric One-Dimensional Constrained Ising Model: Rigorous Results

We study a one-dimensional spin (interacting particle) system, with product Bernoulli (p) stationary distribution, in which a site can flip only when its left neighbor is in state +1. Such models have been studied in physics as simple exemplars of systems exhibiting slow relaxation. In our East model the natural conjecture is that the relaxation time (p), that is 1/(spectral gap), satisfies log (p) as p0. We prove this up to a factor of 2. The upper bound uses the Poincaré comparison argument applied to a wave (long-range) comparison process, which we analyze by probabilistic techniques. Such comparison arguments go back to Holley (1984, 1985). The lower bound, which atypically is not easy, involves construction and analysis of a certain coalescing random jumps process.     

Structure and Photoelectric Characteristics of Functional Polymers and of Compositions Based on Them

The electronic processes in disordered organic solids were analyzed, classified, and summarized. They include the photogeneration of charge pairs, electron transport, geminal and bimolecular recombination, and the capture of charge carriers in local electronic states. General criteria were formulated for the targeted search for and the creation of new prospective polymers and their compositions with organic pigments and related compounds having specific photoelectric characteristics for electrographic and holographic media, photoelectric converters, and electroluminescent and photorefractive devices.     

Conservative Means of Orthogonal Series and the Spaces $$L^p [0;1] $$ , $$p \in (1;\infty ) $$

Necessary and sufficient conditions for an orthogonal series to be the Fourier series of a function in the space , , are obtained. In the special case of regular summation methods we recover the classical results of Orlicz and Lomnicki.     

Dielectric relaxation and ac conductivity of sodium tungsten phosphate glasses

Studies of dielectric relaxation and ac conductivity have been made on three samples of sodium tungsten phosphate glasses over a temperature range of 77–420 K. Complex relative permitivity data have been analyzed using dielectric modulus approach. Conductivity relaxation frequency increases with the increase of temperature. Activation energy for conductivity relaxation has also been evaluated. Measured ac conductivity (σ_m(ω)) has been found to be higher than σ_dc at low temperatures whereas at high temperature σ_m(ω) becomes equal to σ_dc at all frequencies. The ac conductivity obeys the relation σ_ac(ω)=Aω ^S over a considerable range of low temperatures. Values of exponent S are nearly equal to unity at about 78 K and the values decrease non-linearly with the increase of temperature. Values of the number density of states at Fermi level (N(E _F)) have been evaluated at 80 K assuming values of electron wave function decay constant α to be 0.5 (Å)^?1. Values of N(E _F) have the order 10²⁰ which are well within the range suggested for localized states. Present values of N(E _F) are smaller than those for tungsten phosphate glasses.     

Molecular dynamics of polycrystalline cellobiose studied by solid-state NMR

Molecular motions of polycrystalline cellobiose have been investigated by measuring proton spin–lattice relaxation times, T₁ and T_1ρ, and the second moment, M₂, in both protonated and D₂O exchanged forms over the temperature range 120–380 K. T₁ relaxation is dominated by the motions of hydroxyl groups between 150 and 380 K, characterised by an activation energy of about 8.74 kJ/mol, whereas T_1ρ relaxation is driven by the motions of the same groups between 120 and 300 K. T_1ρ results suggest that hydroxyl groups have a distribution of dynamics. Motion of methylene groups was detected in the second-moment experiments at about 350 K, characterised by activation energy of about 40 kJ/mol. Consideration of the calculated and observed rigid-lattice second moments suggests that the reported X-ray data are incorrect for the inter-proton distance on C6′. ¹³C CPMAS spectra of both protonated and deuterated cellobiose have also been measured. Spectra of the deuterated material showed the existence of a second crystalline form in addition to the normal form.