A differential-geometric criterion of the kinematic integrability of nonlinear differential equations

A theorem on the existence of a G-representation and a differential-geometric criterion of the kinematic integrability for nonlinear differential equations from the Λ²-G-classes is proved. Examples of zero-curvature representations and metrics for some equations of mathematical physics are presented. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Runge–Kutta interpolants for high precision computations

Runge–Kutta (RK) pairs furnish approximations of the solution of an initial value problem at discrete points in the interval of integration. Many techniques for enriching these methods with continuous approximations have been proposed. Here we construct C ¹ continuous, eighth and ninth order interpolation methods for a recently appeared RK pair of orders 9(8). These interpolants share a very small leading truncation error making them suitable for use at quadruple precision, i.e. 32–33 decimal digits of accuracy. Extended numerical results justify our effort.     

Circular dichroism spectra in mordant azo dyes

The photogyrotropic properties of several mordant yellow azo dyes in gelatin and polymer matrices that are in the normal state or exposed to circularly polarized light are investigated. A model explanation for the experimental data is given.     

Fabrication of bamboo-shaped GaN nanorods

Bamboo-shaped GaN nanorods were formed through a simple sublimation method. They were characterized by means of X-ray powder diffraction (XRD), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM) and selected-area electron diffraction (SAED). The TEM image showed that the nanorods were bamboo-like. XRD, HRTEM and SAED patterns indicated that the nanorods were single-crystal wurtzite GaN. Received: 8 January 2001 / Accepted: 28 April 2001 / Published online: 20 December 2001     

Scale-free behavior of the Internet global performance

Measurements and data analysis have proved very effective in the study of the Internet's physical fabric and have shown heterogeneities and statistical fluctuations extending over several orders of magnitude. Here we focus on the relationship between the Round-Trip-Time (RTT) and the geographical distance. We define dimensionless variables that contain information on the quality of Internet connections finding that their probability distributions are characterized by a slow power-law decay signalling the presence of scale-free features. These results point out the extreme heterogeneity of Internet delay since the transmission speed between different points of the network exhibits very large fluctuations. The associated scaling exponents appear to have fairly stable values in different data sets and thus define an invariant characteristic of the Internet that might be used in the future as a benchmark of the overall state of “health” of the Internet. Received 25 January 2003 Published online 7 May 2003     

Growth and study of single crystals of (Na,K)NbO<Subscript>3</Subscript> solid solutions

(Na, K)NbO₃ crystals with a perovskite structure and a KNbO₃ content up to 40 mol % were grown from flux with the use of the solvent NaBO₂. The dielectric measurements of the crystals grown revealed phase transitions that had never been observed before in ceramic samples.     

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.