Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

     

Structure of a poly(2,5‐benzimidazole)/phosphoric acid complex

As‐cast films of poly(2,5‐benzimidazole) exhibit uniplanar orientation in which the planes of the aromatic rings lie parallel to the film surface. Upon doping with phosphoric acid, the original crystalline order is lost, but the doped film can be stretched to produce films with uniaxial orientation. After thermal annealing at 540 °C, nine Bragg reflections are resolved in the fiber diagram, and these are indexed by an orthorhombic unit cell with the dimensions a = 18.1 Å, b = 3.5 Å, and c = 11.4 Å, containing four monomer units of two chains. The absence of odd‐order 00l reflections points to a 2₁ chain conformation, which is probably planar so that the aromatic units can be stacked along the b axis. The water and phosphoric acid contents of the crystalline structure cannot be determined exactly because of the presence of extensive amorphous regions that probably have different solvation. The best agreement between the observed and calculated intensities is for an idealized structure containing two phosphoric acids and two water molecules per unit cell. However, the phosphoric acid is probably present mainly in the form of pyrophosphoric acid and its higher oligomers. In addition, the X‐ray data are consistent with a more disordered structure containing chains with random (up and down) polarity and a lack of c‐axis registry. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2576–2585, 2004     

Regularization of some linear ill-posed problems with discretized random noisy data

For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown solution from indirect discrete and noisy data. This procedure is shown to be order optimal for a large class of problems. Smoothness of the solution is measured in terms of general source conditions. The concept of operator monotone functions turns out to be an important tool for the analysis.

     

Bistable dependence of the efficiency of second harmonic generation on the total energy of femtosecond soliton pulses propagating in an optical fiber

Taking into account the effect of cubic nonlinearity on the process of second harmonic generation by soliton-like femtosecond pulses results in a hysteresis dependence of the generation efficiency, the pulse durations, and the mismatch of the wave numbers of the interacting waves on their total energy. Color solitons with earlier unknown shapes are formed at a high efficiency of the frequency conversion. Varying the phase mismatch makes it possible to “switch” the soliton duration and the conversion efficiency without a change in the total energy of the waves or the soliton shape. It is emphasized that the soliton corresponding to the middle branch of the hysteresis dependence is stable against small perturbations. It is significant that the soliton shape remains unchanged in the considered range of energies.     

Investigation of radical decay mechanisms in a glow-discharge plasma in carbon tetrafluoride

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 54, No. 6, pp. 1004–1010, June, 1991.     

Numerical differentiation from a viewpoint of regularization theory

In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

     

H MAS NMR monitoring of the C-labeled carbon scrambling for propane in zeolite H-ZSM-5

It has been demonstrated that ¹H MAS NMR spectroscopy can be used as a tool for in situ monitoring the reaction kinetics of ¹³C-labeled carbon scrambling in alkane molecules adsorbed on zeolite catalysts at the reaction temperature of 540–570 K. The accuracy of the results and the time resolution are improved compared to ¹³C MAS NMR spectroscopy.     

Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log²n) time per output vertex which is an improvement by a factor of O(n/log²n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O(n²⁺) for any positive constant . In the case k<n²⁺, where k is the total size of the input and output, we improve the running to O((n+k+(nk)^2/3)log^O(1)n).