The period function of potential systems of polynomials with real zeros

We consider some analytic behaviors (convexity, monotonicity and number of critical points) of the period function of period annuli of the potential system and focus on the case when g(x) is a polynomial whose roots are all real. The main contributions of this paper are twofold: (i) analytic behaviors are given for the period functions of period annuli surrounding one or more and simple or degenerate equilibria; (ii) as a nontrivial application of the general conclusions in (i), a purely analytical and shorter proof is provided for a result for the case degg=4 recently obtained by Chengzhi Li and Kening Lu with some help of computer algebra [Chengzhi Li, Kening Lu, The period function of hyperelliptic Hamiltonian of degree 5 with real critical points, Nonlinearity 21 (2008) 465-483].     

周期函数Fourier级数展开式的唯一性

以 2 l为周期的函数 f(x)也可看作周期为 2 kl(k=1 ,2 ,3 ,… ) .设 f(x)满足 Dirichlet充分条件 ,[2 ]证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 4l为周期的 Fourier级数对应的不同表达形式是一致的 .本文则在 [2 ]的基础上 ,进一步证明了按 [1 ]方法展开的以 2 l为周期的 Fourier级数和以 2 kl(k=1 ,2 ,3 ,… )为周期的 Fourier级数对应的表达式的一致性 ,从而得出结论 :任一周期函数 f(x)按 [1 ]方法展开的Fourier级数是唯一的 .     

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

The period function of the generalized Lotka-Volterra centers

The present paper deals with the period function of the quadratic centers. In the literature different terminologies are used to classify these centers, but essentially there are four families: Hamiltonian, reversible , codimension four Q₄ and generalized Lotka-Volterra systems . Chicone [C. Chicone, Review in MathSciNet, Ref. 94h:58072] conjectured that the reversible centers have at most two critical periods, and that the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, only the monotonicity of the Hamiltonian and Q₄ families [W.A. Coppel, L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993) 1357-1365; Y. Zhao, The monotonicity of period function for codimension four quadratic system Q₄, J. Differential Equations 185 (2002) 370-387] has been proved. Concerning the family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka-Volterra centers [F. Rothe, The periods of the Volterra-Lokta system, J. Reine Angew. Math. 355 (1985) 129-138; R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984) 1-31; J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986) 178-184]. By means of the first period constant one can easily conclude that the period function of the centers in the family is monotone increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone's conjecture, it should be also monotone increasing near the outer boundary, which in the Poincaré disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there is no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.     

A period vehicle routing case study

The period vehicle routing problem is a multilevel problem assembling two classical problems: the assignment problem and the vehicle routing problem. Collection days have to be assigned to each customer and vehicle routes have to be designed for each day of the period (time horizon) so that the total distribution cost is minimised. The interaction between the temporal and spatial aspects turns the problem into one of the most challenging variations of vehicle routing. In this paper, we present the study of a real period vehicle routing system: the collection of recycling paper containers in the City Council of Almada, Portugal.     

基于群智能算法的预防性维修周期优化

分析将蚁群优化算法应用于预防性维修周期工程寻优问题时遇到的算法参数选择困难等问题,提出将粒子群优化算法和空间划分方法引入该过程以改进原蚁群算法的寻优规则和历程.建立混合粒子群和蚁群算法的群智能优化策略:PS_ACO(Particle Swarm and Ant Colony Optimization),并将其应用于混联系统预防性维修周期优化过程中,以解决由于蚁群算法中参数选择不当和随机产生维修周期解值带来的求解精度差、寻优效率低等问题.算法的寻优结果对比分析表明:该PS_ACO算法应用于预防性维修周期优化问题,在寻优效率及寻优精度上有部分改进,且可相对削弱算法参数选择对优化结果的影响.     

The cyclicity and period function of a class of quadratic reversible Lotka-Volterra system of genus one

In this paper, we investigate a class of quadratic reversible Lotka-Volterra system of genus one with b=3/5. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. Moreover, we prove that the period function of its period trajectories is monotone increasing.     

Period-doubling cascades for large perturbations of Hénon families

The Hénon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur [5].     

基于水量水质耦合多目标模型的辽源地区水资源优化配置

对水资源进行优化配置是解决社会经济发展与水资源可用量紧张的有效手段.采用区域水质—水量耦合水资源优化配置模型,以吉林省辽源市2010年数据为基准,对2020年水资源配置进行优化预测.研究结果表明,辽源市在规划期内"三生"用水结构由89:10:1调整到81:18:1.其中生产用水中第一产业用水量下降2.17%,第二产业用水总量下降2.78%.第三产业用水量提升4.29%,同时降低了水污染的排放.优化方案可有效降低辽源市水资源消耗与水环境污染,为水资源的可持续利用提供了有效的技术支持.     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

Busy period analysis of the level dependent PH/PH/1/K queue

In this paper, we study the transient behavior of a level dependent single server queuing system with a waiting room of finite size during the busy period. The focus is on the level dependent PH/PH/1/K queue. We derive in closed form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. We differentiate between two types of losses: the overflow losses that are due to a full queue and the losses due to an admission controller. For the M/PH/1/K, M/PH/1/K under a threshold policy, and PH/M/1/K queues, we determine simple expressions for their joint transforms.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Optimal control of a dam under seasonal electricity prices

This study concerns the optimal control of a hydroelectric dam under seasonal electricity prices: high in winter, low in summer. The goal is to maximize the expected discounted infinite-horizon return through a policy that determines the amount of electricity to be produced in each time period, depending on the water level at the beginning of the period under consideration. The prices are assumed to be deterministic, and the flows into the reservoir are seasonal, stochastic, but independent from one period to another. The electric power generated is proportional to the amount of water flow through the turbines. There exist seepage and evaporation losses.It is shown that in the simplest price structure, the optimal policy is entirely determined by a single critical water level in each period of time, at which one starts producing. An example shows that the discretization of the reservoir levels can destroy this property. A method is proposed to avoid this difficulty. Another way of defining a policy is through goal-levels. This approach is shown to give higher returns than the standard approach.     

Finite horizon approximations of infinite horizon linear programs

This paper describes the class of infinite horizon linear programs that have finite optimal values. A sequence of finite horizon (T period) problems is shown to approximate the infinite horizon problems in the following sense: the optimal values of theT period problems converge monotonically to the optimal value of the infinite problem and the limit of any convergent subsequence of initialT period optimal decisions is an optimal decision for the infinite horizon problem.     

Control limits for two-state partially observable Markov decision processes

A production process can be in either a GOOD or a BAD state. The true state is unknown and can only be inferred from observations. If the state is good during one period it may deteriorate and become bad during the next period. Two actions are available: CONTINUE or REPLACE (for a fixed cost). The objective is to maximize the expected discounted value of the total future profits.     

我国人口发展总量的中长期预测模型

通过对1994―2005年男、女出生人口性别比的变化和2001―2005年城镇化趋势的分析,建立了我国人口发展总量的中长期预测模型,并针对在总和生育率四种不同情况下2006-2100年的人口发展状况进行了长期预测和详细分析.在目前人口结构状况下,将总和生育率控制在更替水平左右,既可保证总人口不超过15亿,又能降低人口老龄化程度,可为人口控制、决策提供较为科学的依据.     

Fibonacci数列的模数列的周期的一个性质

Fibonacci数列的模数列是周期数列,并且是纯周期数列.利用模数列的定义,讨论了Fibonacci数列的模数列的周期的一个性质,证明了下列结果:假设m1与m2为不同的正整数,Fibonacci数列{Fn}的模数列{an(m1)}与{an(m2)}的最小正周期分别为T1与T2,则模数列{an([m1,m2])}的最小正周期为[T1,T2].     

Calculation of effective mechanical properties of textiles by asymptotic approach

We consider plates with 2-D periodic rod or fabric structure, used as geotextiles or textiles. The period of structure as well as the hight of a plate are much smaller compared to its depth and width. This makes a direct numerical computation of boundary value or contact elasticity problem too expensive. Three small parameters are introduced for the asymptotic analysis: the first one connected with the period of structure, the second one with the thickness of fibers or beams inside the periodicity cell and the last one – with the hight of a plate. The overcoming to the limit with respect to period of structure provides equivalent homogenized plate of the finite hight. Calculation of its outer-plane stiffness is a new aspect of this work. The next overcoming to the limit with respect to the hight reduces the 3-D problem to the homogenized equations, fourth order PDEs. The effective elasticity moduli and outer-plane stiffness can be obtained numerically solving cell experiments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A direct approach to sojourn times in a busy period of an M/M/1 queue

In this paper we present a direct approach to obtaining joint distributions of various quantities of interest in a busy period in an M/M/1 queue. These quantities are: the sojourn times and waiting times of all the customers in the busy period, the busy period length and the number of customers served in a busy period. Since the evolution of the total workload process between two successive customer arrivals is deterministic, this work gives statistic of the complete evolution of the workload process within a busy period. This work was done when the author was post doctoral fellow with the MAESTRO group at INRIA, Sophia Antipolis, France, and was supported by project no. 2900-IT-1 from the Centre Franco-Indien pour la Promotion de la Recherche Avancee (CEFIPRA).     

An EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment

In this paper, an EOQ (Economic Order Quantity) model is developed for a deteriorating item having time dependent demand when delay in payment is permissible. The deterioration rate is assumed to be constant and the time varying demand rate is taken to be a quadratic function of time. Mathematical models are also derived under two different circumstances, i.e. Case I: The credit period is less than or equal to the cycle time for settling the account and Case II: The credit period is greater than the cycle time for settling the account. The results are illustrated with numerical examples. Justification for considering a time quadratic demand and permissible delay in payment are discussed.