Geometric integration methods for general nonlinear dynamic equation based on Magnus and Fer expansions

Based on Magnus or Fer expansion for solving linear differential equation and operator semi-group theory, Lie group integration methods for general nonlinear dynamic equation are studied. Approximate schemes of Magnus type of 4th, 6th and 8th order are constructed which involve only 1, 4 and 10 different commutators, and the time-symmetry properties of the schemes are proved. In the meantime, the integration methods based on Fer expansion are presented. Then by connecting the Fer expansion methods with Magnus expansion methods some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed. These methods belong to the category of geometric integration methods and can preserve many qualitative properties of the original dynamic system.     

Research of optimization to solve nonlinear equation based on granular computing

In this article, a real number is defined as a granulation and the real space is transformed into real granu-lar space[1]. In the entironment, solution of nonlinear equation is denoted by granulation in real granular space. Hence,the research of whole optimization to solve nonlinear equation based on granular computing is proposed[2]. In classicalcase, we solve usually accurate solution of problems. If can't get accurate solution, also finding out an approximate solutionto close to accurate solution. But in real space, approximate solution to close to accurate solution is very vague concept. Inreal granular space, all of the approximate solutions to close to accurate solution are constructed a set, it is a granulation inreal granular space. Hence, this granulation is an accurate solution to solve problem in some sense, such, we avoid to sayvaguely "approximate solution to close to accurate solution". We introduce the concept of granulation in one dimension real space. Any positive real number a together with movinginfinite small distance ε will be constructed an interval [a-ε,a ε], we call it as granulation in real granular space, denotedby ε(a) or [a]. We will discuss related properties and operations[3] of the granulations. Let one dimension real space be R, where each real number a will be generated a granulation, hence we get a granularspace R* based on real space R. Obviously, R∈R*. Infinite small number in real space R is only O, and there are three in-finite small granulations in real number granular space R* : [0], [ε] and [-ε]. As the graph in Fig. 1 shows. In Fig. 1,[-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation,[0] is a infinite small granulation.[a] is a granulation of real number a generating, it could be denoted by interval [a-ε,a ε] in real space [3-5].Letf(x)=0 be a nonliner equation,its graph in interval[-3,10]id showed in Fig.2.Where -3≤x≤10 Relation ρ(f‖,ε)is defied is follows:(x1,x2)∈ p(f‖,ε)iff |f(x1)- f(x2)|<εWhere ε is any given small real number.We have five appoximate solution sets on the nonliner equation f(x)=0 by ρ(f‖,ε)∧|f(x)|[a,b]max,to denote by granulations[xi1 xi2/2],[xi3 xi4/2],[xi5 xi6/2],[xi7 xi8/2]and[xi9 xi10/2]respectively,where |f(x)|[a,b]max denotes local maximum on x ∈[a,b].This is whole optimum on nonliear equation in interval [-3,10].We will get best opmension solution on nonliner equation via computing f(x)to use the five solutions dented by grandlation in one dimension real granlar space[2,5].     

FluxExplorer： A general platform for modeling and analyses of metabolic networks based on stoichiometry

An increasing number of genomic and biochemical data make it possible to reconstruct biochemical net-works, especially metabolic networks, of an organelle or even a whole cell. Some methods for metabolism modeling and analyses in this field have been deve…     

Subharmonic and ultraharmonic emissions based on the nonlinear oscillation of encapsulated microbubbles in ultrasound contrast agents

Subbarmonics or ultraharmonics provides better contrast-to-tissue ratio （CTR） than the fundamental or the second harmonics, having prospective application in medical diagnosis. In this paper, subharmonic and ultraharmonic emissions are theoretically studied through nonlinear oscillation of encapsulated bubbles. The optimized frequencies for emissions of the subharmonics and ultraharmonics are discussed. In addition, sound pressure dependences of the subharmonics and ultraharmonics are studied in theory as well as in measurement. Results reveal that the developments of both subharmonics and ultraharmonics have the same trend, i.e. occurrence, growth and saturation, but the generation of ultraharmonic is a little earlier than that of subharmonic.     

Friction compensation design based on state observer and adaptive law for high-accuracy positioning system

Friction is one of the main factors that affect the positioning accuracy of motion system. Friction compensation based on friction model is usually adopted to eliminate the nonlinear effect of friction. This paper presents a proportional-plus-derivative (PD) feedback controller with a friction compensator based on LuGre friction model. We also design a state observer to observe the unknown state of LuGre friction model, and adopt a parameter adaptive law and off-line approximation to estimate the parameters of LuGre friction model. Comparative experiments are carried out among our proposed controller, PD controller with friction compensation based on classical friction model, and PD controller without friction compensation. Experimental results demonstrate that our proposed controller can achieve better performance, especially higher positioning accuracy.     

Estimating genetic diversity and sampling strategy for a wild soybean （Glycine soja） population based on different molecular markers

With the rapid development of the global economy and continued increase in world population, natural environments face serious deterioration and change, which has led to the extinction or severe endangerment of many plant species including important crops…