scholarly journals Higher order stroboscopic averaged functions: a general relationship with Melnikov functions

2021 ◽  
pp. 1-9
Author(s):  
Douglas Novaes
Keyword(s):  
Differential Equation ◽  
Averaging Method ◽  
Higher Order ◽  
Research Literature ◽  
General Relationship ◽  
The Other ◽  
Asymptotic Estimates ◽  
Melnikov Functions ◽  
Averaging Functions ◽  
Identity Transformations

In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous T-periodic differential equations of the kind x′=ε F(t,x,ε ). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi's which controls in some sense the existence of isolated T-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincaré–Pontryagin–Melnikov functions or just Melnikov functions. While it is known that f1=Tg1, a general relationship between gi and fi is not known so far for i≥ 2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.

scholarly journals Voigt Transform and Umbral Image

2020 ◽  
Vol 25 (3) ◽  
pp. 49
Author(s):  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Marcello Artioli ◽  
Giuseppe Dattoli
Keyword(s):  
Spectroscopic Studies ◽  
Higher Order ◽  
Point Of View ◽  
The Other ◽  
Pivotal Role ◽  
Umbral Calculus ◽  
Laguerre Functions ◽  
Hermite And Laguerre Functions ◽  
Higher Order Functions ◽  
Voigt Function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

scholarly journals New Properties of Modified Second Kind Chebyshev Polynomials

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Semaa Hassan Aziz ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Digital Computer ◽  
Higher Order ◽  
Algebraic Equations ◽  
Equation Problem ◽  
Higher Order Differential Equations

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

scholarly journals A Study of Nonlinear Boundary Value Problem

2021 ◽  
Author(s):  
Noureddine Bouteraa ◽  
Habib Djourdem
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Nonlinear Boundary Value Problem ◽  
Point Boundary ◽  
Error Estimations ◽  
Fractional Differential

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

A memory system in the monkey

1982 ◽  
Vol 298 (1089) ◽  
pp. 85-95 ◽  
Keyword(s):  
Neural Model ◽  
Memory System ◽  
Higher Order ◽  
The Other ◽  
Dorsomedial Nucleus ◽  
Memory Processes ◽  
Stimulus Representation ◽  
Parallel Circuits ◽  
Sensory Pathway ◽  
Cortical Circuit

A neural model is presented, based largely on evidence from studies in monkeys, postulating that coded representations of stimuli are stored in the higher-order sensory (i.e. association) areas of the cortex whenever stimulus activation of these areas also triggers a cortico-limbo-thalamo-cortical circuit. This circuit, which could act as either an imprinting or rehearsal mechanism, may actually consist of two parallel circuits, one involving the amygdala and the dorsomedial nucleus of the thalamus, and the other the hippocampus and the anterior nuclei. The stimulus representation stored in cortex by action of these circuits is seen as mediating three different memory processes: recognition, which occurs when the stored representation is reactivated via the original sensory pathway; recall, when it is reactivated via any other pathway; and association, when it activates other stored representations (sensory, affective, spatial, motor) via the outputs of the higher-order sensory areas to the relevant structures.

Universals, innateness and explanation in second language acquisition

2004 ◽  
Vol 28 (3) ◽  
pp. 682-703 ◽  
Author(s):  
Fred Eckman
Keyword(s):  
Second Language ◽  
Second Language Acquisition ◽  
Language Acquisition ◽  
Universal Grammar ◽  
Higher Order ◽  
The Other ◽  
Language Typology ◽  
Domain Specific ◽  
Other Hand ◽  
Typological Approach

This paper considers the question of explanation in second language acquisition within the context of two approaches to universals, Universal Grammar and language typology. After briefly discussing the logic of explaining facts by including them under general laws (Hempel & Oppenheim 1948), the paper makes a case for the typological approach to explanation being the more fruitful, in that it allows more readily for the possibility of ‘explanatory ascent’, the ability to propose more general, higher order explanations by having lower-level generalizations follow from more general principles. The UG approach, on the other hand is less capable of such explanatory ascent because of the postulation that the innate, domain-specific principles of UG are not reducible in any interesting way to higher order principles of cognition (Chomsky 1982).

scholarly journals Field Experiments in Class Size from the Early Twentieth Century

2009 ◽  
Vol 23 (4) ◽  
pp. 211-230 ◽  
Author(s):  
Jonah Rockoff
Keyword(s):  
World War Ii ◽  
Public Schools ◽  
Class Size ◽  
Large Scale ◽  
Field Experiments ◽  
Research Literature ◽  
The Other ◽  
World War ◽  
Mixed Messages

A vast majority of adults believe that class size reductions are a good way to improve the quality of public schools. Reviews of the research literature, on the other hand, have provided mixed messages on the degree to which class size matters for student achievement. Here I will discuss a substantial, but overlooked, body of experimental work on class size that developed prior to World War II. These field experiments did not have the benefit of modern econometrics, and only a few were done on a reasonably large scale. However, they often used careful empirical designs, and the collective magnitude of this body of work is considerable. Moreover, this research produced little evidence to suggest that students learn more in smaller classes, which stands in contrast to some, though not all, of the most recent work by economists. In this essay, I provide an overview of the scope and breadth of the field experiments in class size conducted prior to World War II, the motivations behind them, and how their experimental designs were crafted to deal with perceived sources of bias. I discuss how one might interpret the findings of these early experimental results alongside more recent research.

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