A gradient approach to the determination of stability of limit cycles

This paper presents the applications of the stability equation method to the analysis and design of nonlinear systems. A simplified criterion for finding the limit cycle in a parameter plane is presented, and a simplified method for testing limit cycle stability is introduced. Several examples, considered in the current literature, are used for illustrating the advantages of the method presented.     

Theory of nonlinear systems

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input.Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u₁, u₂,…, u_s, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get $\text{c(t)V(α)}\text{=}\text{A}{}_{\mathrm{\alpha }}\text{2л}{}^{\mathrm{<mtext>s</mtext><mtext>2</mtext>}}$, where the upper bar denotes time average and A_α's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of A_α's.The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems.From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φ_xy can be shown to be equal to the convolution of system impulse response h₁ with the autocorrelation φ_xx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φ_xy(σ) = Ah₁(σ), while the autocorrelation is φ_xx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry.Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results.A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.     

Couple-stress Effects Near an Interior Hole of an Infinite Elastic Plane Subjected to a Concentrated Force

Couple-stress solutions are obtained near the hole boundary of an infinite elastic plane subjected to a radial concentrated force at x = b, y = 0,under the restriction of r/b<-0.2 and b/l?;-10. Here r and l are the distance from the center of the hole to the point of interest and a characteristic length (couple-stress constant). At r = a, θ = ±π2, the stress ratio of the couple-stress solution to the classical one for the current problem is slightly lower than one of the Mindlin problem (1) for some values of v, except for v = 0.5 where they are equal. The dimensionless stress σ&#x0302;_θθ is found along the hole boundary.     

Antihyperglycemic effect of water extract of dry fruits of<Emphasis Type="Italic">Terminalia chebula</Emphasis> in experimental diabetes mellitus

Water extract of dry fruits ofTerminalia chebula (Hindi-Harda, Telugu-Karakkaya) at a dose of 200 mg/kg body weight improved the glucose tolerance as indicated by 44% of reduction in the peak blood glucose at 2^nd hour in glucose tolerance test in diabetic (streptozotocin induced) rats. Treatment of diabetic rats with an initial fasting blood glucose of 253±9.4 mg/dl daily once with the water extract (200 mg/kg) for two weeks brought down the fasting blood glucose to 123±8.4 mg/dl which is only slightly above the normal value. These results indicate that water extract of Terminalia chebula improves glucose tolerance and brings down fasting blood glucose in diabetic rats.     

Analysis of Nonlinear Systems with Stochastic Input and Stochastic Parameters

This paper gives an extension of the Volterra-functional method to the analysis of nonlinear systems with both stochastic input and stochastic parameters. In Section II, first-order linear systems with stochastic input and stochastic parameter are analyzed. Certain definitions and symbolic relations are established. A special example in which the stochastic input and the stochastic parameter are closely related is given in Appendix I. Section III deals with the analysis of first-order nonlinear systems with both stochastic input and stochastic parameter, by means of the Volterra-functional method. A simpler analysis by extension of the Transform-ensemble method is given in Appendix II. Section IV presents the detailed analysis of nonlinear stochastic systems with four illustrative examples. The analysis can be extended to a class of nonlinear stochastic systems with higher-order linear plants with constant or time-varying parameters.     

The composition of and formative influences on the curriculum

Up until the last two decades the design of the school curriculum was mainly based on empirical foundations. This often involved arbitrary decisions as to what to select from mankind's past and present experience as curriculum content. This problem became especially acute as a result of school reforms in all parts of the world in the 1960–1975 period. Since then the main task of theoretical work in didactics in the curriculum area has been to remove this arbitrariness and throughly to investigate the composition of and formative influences on the curriculum.

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Zusammenfassung Bis in die letzten zwei Jahrzehnte wurde das Schulcurriculum hauptsächlich empirisch gestaltet. Dies führte oft zu willkürlichen Entscheidungen darüber, welche Erfahrungen aus Vergangenheit und Gegenwart für das Curriculum ausgewählt werden sollten. Dieses Problem wurde nach den Schulreformen in aller Welt zwischen 1960 und 1975 besonders akut. Seitdem besteht die Hauptaufgabe der theoretischen didaktischen Arbeit auf dem Gebiet des Curriculums darin, diese Willkür absuzchaffen und die wesentlichen Einflüsse auf das Curriculum und dessen Zusammensetzung gründlich zu untersuchen.

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Résumé L'élaboration des programmes scolaires était fondée sur des bases empiriques jusqu'aux deux dernières décennies. Cette situation se caractérisait souvent par des décisions arbitraires quant aux éléments du passé de l'humanité et de l'expérience présente qui pouvaient être intégrés dans le contenu du curriculum. Ce problème s'aiguisa particulièrement à la suite des réformes scolaires introduites dans toutes les régions du monde dans les années 1960–1975. Dès lors, dans le domaine du curriculum, la tâche majeure du travail théorique en matière de didactique a consisté à éliminer cet arbitraire et à étudier minutieusement la composition du curriculum et les influences formatives exercées sur celuici.

     

Longitudinal Trajectories,Social and Individual Antecedents,and Outcomes of Problematic Internet Use Among Late Adolescents

Given the detrimental effects associated with problematic internet use (PIU) and the need to better understand its nature and evolution, the present study examined the development of PIU in a sample of 1,750 adolescents (aged 16–19) from Finland over a 3-year period. We documented the social (loneliness, perceived maternal and paternal behaviors) and individual (sex) antecedents, as well as the outcome implications (depressive symptoms, substance use, academic achievement) of PIU trajectories. Outcomes also predicted PIU trajectories. Latent curve modeling revealed an initially moderate, and subsequently decreasing trajectory of PIU. PIU was predicted by loneliness, paternal neglect, maternal care, depressive symptoms, and being male. In turn, PIU trajectories predicted increases in depressive symptoms and substance use, but decreases in academic achievement.     

Are we all in this together?: mathematics teachers’ perspectives on equity in remote instruction during pandemic

Educational Studies in Mathematics - The COVID-19 pandemic is an unprecedented situation that influenced all aspects of society, including education. Millions of students found themselves adjusting...