A Note on a Functional Differential Equation with State Dependent Argument

This paper is concerned with solutions of a functional differential equation. Using Krasnoselskii’s fixed point theorem, the solutions can be obtained from periodic solutions of a companion equation.

Quantifying glacial elevation changes in the central Qilian Mountains during the early 21^(st) century

Glaciers in the central Qilian Mountains provide important water resources for the arid Hexi corridor and Qaidam Basin;however,changes in these glaciers interact with climate change.Twenty-four bi-static image pairs of TerraSAR-X add-on for Digital Elevation Measurement(TanDEM-X)data,in addition to a Shuttle Radar Topography Mission-C/X band digital elevation model,and the technology of iterative differential synthetic aperture radar interferometry were used to carry out glacier elevation change analysis in the central Qilian Mountains in China during 2000–2014.Glacier elevation changed with an average rate of(−0.47±0.06)m yr^(−1),while changes in elevation of(−0.51±0.06)m yr^(−1) and(−0.44±0.06)m yr^(−1) were found in the northern(including the Zoulangnan,Tuolai,and Tuolainan mountains)and southern(including the Shulenan and Hark mountains)regions,respectively.Summer mean temperature has risen by 0.51℃(10 yr)^(−1)in the northern region and 0.48℃(10 yr)^(−1) in the southern region during 1989–2014;however,the change in amplitude of annual precipitation was 2.69 mm yr^(−1) in the northern region and 4.77 mm yr^(−1) in the southern region for the same period.These changes can be ascribed as major driving factors for the differences in the changes in glacial elevation in the northern and southern regions.Four types of glaciers existed in the region when considering the change in elevation of the glacial tongue and variation in the position of the glacial terminus:surging,advancing,intensively retreating,and slightly retreating glaciers.If elevation decreased more than 20 m on the part of glacier tongue,the glacier terminus position had commonly retreated more than 100 m.

Analytic Solutions of an Iterative Differential Equation under Brjuno Condition

In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown functionαy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.

On a novel tracking differentiator design based on iterative learning in a moving window

Differential signals are key in control engineering as they anticipate future behavior of process variables and therefore are critical in formulating control laws such as proportional-integral-derivative(PID).The practical challenge,however,is to extract such signals from noisy measurements and this difficulty is addressed first by J.Han in the form of linear and nonlinear tracking differentiator(TD).While improvements were made,TD did not completely resolve the conflict between the noise sensitivity and the accuracy and timeliness of the differentiation.The two approaches proposed in this paper start with the basic linear TD,but apply iterative learning mechanism to the historical data in a moving window(MW),to form two new iterative learning tracking differentiators(IL-TD):one is a parallel IL-TD using an iterative ladder network structure which is implementable in analog circuits;the other a serial IL-TD which is implementable digitally on any computer platform.Both algorithms are validated in simulations which show that the proposed two IL-TDs have better tracking differentiation and de-noise performance compared to the existing linear TD.

ANALYTIC SOLUTIONS OF AN ITERATIVE EQUATION WITH FIRST ORDER DERIVATIVE

This paper is concerned with a nonlinear iterative equation with first order derivative. By construction a convergent power series solution, analytic solutions for the original equation are obtained.