A study on functional independence of the Iwasawa power series

We study a class of functional independences that the Iwasawa power series satisfy for both zero and non-zero characteristics. As results, we prove a generalization of Anglès and Ranieri [B. Anglès, G. Ranieri, On the linear independence of p$p$-adic L$L$-functions modulo p$p$, Ann. Inst. Fourier (Grenoble) 60 (5) (2010) 1831–1855] and transcendence of the Iwasawa power series over the rational functions for non-zero characteristics. We also verify that the power series is not a solution of any non-trivial linear differential equation with the coefficients of rational functions over p$p$-adic numbers.     

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

Summary We have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.     

A kinetic study of the reaction of diols with hydrogen halides

The kinetics of the reaction of diols with various hydrogen halides is discussed. The reaction involves neighbouring group participation as postulated earlier, the rate of substitution being faster for a heavier nucleophile. The reactions are inert in aqueous systems as the X^? nucleophilicity becomes negligible as compared to the nucleophilicity of water. At and above 7·5 per cent. water these reactions do not occur. Variation of dielectric constant also notably affects the process.     

A generalization of the Laurent series

A system of functions, biorthogonal to the D-basis introduced by M. K. Fage for a linear differential operator D with analytic coefficients, is constructed. Series in the combination of the D-basis with this system generalize Laurent series in the same way as series in the D-basis generalize Taylor series.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 521–531, May, 1971.     

A study of convective difussion of admixtures in a medium with traps using orthogonal series

We construct a generalized Laguerre transform and establish some of its properties. We illustrate the application of this transform with use of exponential Fourier series to solve a heterodiffusion problem.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 64–70.     

A generalization of the Gauss hypergeometric series

A generalization of the Gauss hypergeometric function to t variables is given, and the Euler identity is shown to hold for this generalized function. The corresponding generalization of the Saalschütz theorem is also obtained.     

Numerical study of the controlled Van der Pol oscillator in Chebyshev series

Summary The optimal control of the Van der Pol oscillator is investigated by a numerical method based on the expansion of the state function and the control strategy in Chebyshev series. The optimal control problem is reduced to a parameter optimization problem.

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Resumé On étudie le contrôle optimal de l'oscillateur de Van der Pol à l'aide d'une méthode numérique qui utilise le développement des fonctions de position et de contrôle en série de Chebyshev. Le problème de contrôle optimal est réduit à un problème d'optimisation de paramètres.

     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

A series of coverings of the regular n-gon

We define an infinite series of translation coverings of Veech’s double-n-gon for odd n ≥ 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n ≥ 8. These families give rise to explicit examples of infinite translation surfaces with lattice Veech group.     

A method of summation of the Fourier-Jacobi series

A summation method is constructed for the Fourier-Jacobi series, which has properties similar to the properties of the de la Vallée-Poussin methods of summation of the Fourier series by the trigonometric system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 676–680, May, 1993.