关于几乎处处连续的本性函数的可积性问题

以勒贝格可测函数与几乎处处连续的本性函数几乎处处相等及零集上的积分等于零为前提,按照继承性,可求性,收敛性原则定义[a,b]上几乎处处连续的本性函数的积分,引进一致局部可积与无穷断度点上积分一致收敛概念,给出函数可积的充要条件.

关于本性函数积分定义的注

对照几乎处处连续的本性函数的积分定义,给出几乎处处连续的本性函数绝对可积的充要条件,证明几乎处处连续的本性函数积分是非绝对可积的,将无穷区间上几乎处处连续的本性函数的积分问题化为有穷区间的积分。

本性平方函数在Campanato空间上的存在性和有界性

令S_α(f)是f的本性Lusin平方函数.若f属于Campanato空间f∈L^(p,β),1<p<∞,-n/p≤β<1,我们证明了,若存在一点x_0∈R^n,使得S_α(f)(x_0)<∞,则S_α(f)(x)在Rn上几乎处处有限,且存在常数C,使得‖S_α(f)‖_(Lp,β)≤C‖f‖_(Lp,β).类似结论对本性Littlewood-Paley g-函数也成立.

收敛函数列的一个性质

给出函数间断度定义、本性间断点定义及几乎处处连续的本性函数定义,由勒贝格可测函数的本性定理将收敛的几乎处处连续的本性函数列的上、下确界函数本性化,证明收敛的几乎处处连续的本性函数列的无界点集的闭包S∞为零集.

一类Littlewood-Paley g_λ~*-函数在Campanato空间上的有界性

建立了一类Littlewood-Paley g_λ~*-函数在Campanato空间上的有界性.并证明了若上述函数在一点有限,则其在Rn上几乎处处有限,且在Campanato空间上是有界的.

关于本性最大模的一点注记

本文给出本性最大模的几个等价定义。

关于积分与极限交换的充要条件

由几乎处处收敛的几乎处处连续的本性函数序列的无界点集的闭包S∞是零集,结合几乎处处连续的本性函数的积分定义与可积的充要条件,证明几乎处处收敛的几乎处处连续的本性函数序列的积分与极限交换的充要条件是该函数列在S∞上积分等度收敛.

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Diracequation.The analytical solution for the Makarov potential with pseudospin symmetry is obtained by Nikiforov-Uvarov(N-U)method.The eigenfunctions and eigenenergies are presented with equal mixture of vector and scalar potentials inopposite signs,for which is exact.

关于Kantorovich型不等式

给出了一种积分形式的Kantorovich型不等式为 :设a,A ,b,B和α均为正数 ,且a<A ,b <B .设E是可测集且μ(E) <+∞ .若p是一个在E上几乎处处为正的可积函数 ,f和g是在E上几乎处处为正的可测函数 ,且几乎处处有a≤f≤A ,b≤g≤B ,则∫Epfαdμ∫Epgαdμ ∫Ep(fg) α2 dμ2 ≤ 14ABabα4+ abABα42同时建立了等号成立的条件 .

线性同构下有界函数空间的球覆盖性质

赋范线性空间称为满足球覆盖性质,若其单位球面可被可数个不包含原点的闭球覆盖.考虑了C_(b)(R)和L^(∞)(R)两个例子上的等价范数|·|_(a)和‖·‖_(λ),得到两个结果:(C_(b)(R),|·|_(a))满足足球覆盖性质当且仅当a∈(1/2,1];(L^(∞)(R),‖·‖_(λ))在λ∈[0,1]时均不满足足球覆盖性质.

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.

Spectral-domain dyadic Green’s functions in chiral media

A novel method of deriving the electromagnetic dyadic Green＇s functions in an unbounded, lossless, reciprocal and homogeneous chiral media described by the constitutive relations D = εE ＋ jγB and H = jγE ＋ μ^-1B - （ωε）^-1γJ is given. The divergenceless and irrotational splitting of dyadic Dirac 8 function and Fourier transformation are used to directly obtain the divergenceless and irrotational component of spectral-domain dyadic Green＇s functions in chiral media. This method avoids using the dyadic Green＇s function eigenfunction expansion technique. The method given here can be generalized to a source-free region and an achiral case.

The Chaotic Monopoly Price Growth Model

Deterministic chaos refers to an irregular or chaotic motion that is generated by nonlinear systems. The chaotic behavior is not to quantum-mechanical-like uncertainty. Chaos theory is used to prove that erratic and chaotic fluctuations can indeed arise in completely deterministic models. Chaotic systems exhibit a sensitive dependence on initial conditions. Seemingly insignificant changes in the initial conditions produce large differences in outcomes. To maximize profit, the monopolist must first determine its costs and the characteristics of market demand. Given this knowledge, the monopoly firm must then decide how much to produce. The monopoly firm can determine price, and the quantity it will sell at that price follows from the market demand curve. The basic aim of this paper is to construct a relatively simple chaotic growth model of the monopoly price that is capable of generating stable equilibria, cycles, or chaos. A key hypothesis of this work is based on the idea that the coefficient,π=[m（a-1）（e-1）^-eb]plays a crucial role in explaining local stability of the monopoly price, where,b^the coefficient of the marginal cost function of the monopoly firm, m--the coefficient of the inverse demand function, e--the coefficient of the price elasticity of the monopoly demand, a--the coefficient.

Multi-objective nonlinear model predictive control through switching cost functions and its applications to chemical processes

This paper proposes a switching multi-objective model predictive control(MOMPC) algorithm for constrained nonlinear continuous-time process systems.Different cost functions to be minimized in MPC are switched to satisfy different performance criteria imposed at different sampling times.In order to ensure recursive feasibility of the switching MOMPC and stability of the resulted closed-loop system,the dual-mode control method is used to design the switching MOMPC controller.In this method,a local control law with some free-parameters is constructed using the control Lyapunov function technique to enlarge the terminal state set of MOMPC.The correction term is computed if the states are out of the terminal set and the free-parameters of the local control law are computed if the states are in the terminal set.The recursive feasibility of the MOMPC and stability of the resulted closed-loop system are established in the presence of constraints and arbitrary switches between cost functions.Finally,implementation of the switching MOMPC controller is demonstrated with a chemical process example for the continuous stirred tank reactor.

Unit Commitment with Production Cost Uncertainty： A Recourse Programming Method

Many studies have considered the solution of Unit Commitment problems for the management of energy networks. In this field, earlier work addressed the problem in determinist cases and in cases dealing with demand uncertainties. In this paper, the authors develop a method to deal with uncertainties related to the cost function. Indeed, such uncertainties often occur in energy networks （waste incinerator with a priori unknown waste amounts, cogeneration plant with uncertainty of the sold electricity price...）. The corresponding optimization problems are large scale stochastic non-linear mixed integer problems. The developed solution method is a recourse based programming one. The main idea is to consider that amounts of energy to produce can be slightly adapted in real time, whereas the on/off statuses of units have to be decided very early in the management procedure. Results show that the proposed approach remains compatible with existing Unit Commitment programming methods and presents an obvious interest with reasonable computing loads.

A singularity-based eigenfunction decomposition for Kohn-Sham equations

We show that the eigenfunctions of Kohn-Sham equations can be decomposed as ■ = F ψ, where F depends on the Coulomb potential only and is locally Lipschitz, while ψ has better regularity than ■.

Squared Eigenfunction Symmetries for the BTL and CTL Hierarchies

In this paper, the squared eigenfunction symmetries for the BTL and CTL hierarchies are expficitly constructed with the suitable modification of the ones for the TL hierarchy, by considering the BTL and GTL constraints. Also the connections with the corresponding additional symmetries are investigated： the squared eigenfunction symmetry generated by the wave function can be viewed as the generating function for the additional symmetries.

A learning method for energy optimization of the plug-in hybrid electric bus

The optimal energy management for a plug-in hybrid electric bus(PHEB)running along the fixed city bus route is an important technique to improve the vehicles’fuel economy and reduce the bus emission.Considering the inherently high regularities of the fixed bus routes,the continuous state Markov decision process(MDP)is adopted to describe a cost function as total gas and electric consumption fee.Then a learning algorithm is proposed to construct such a MDP model without knowing the all parameters of the MDP.Next,fitted value iteration algorithm is given to approximate the cost function,and linear regression is used in this fitted value iteration.Simulation results show that this approach is feasible in searching for the control strategy of PHEB.Simultaneously this method has its own advantage comparing with the CDCS mode.Furthermore,a test based on a real PHEB was carried out to verify the applicable of the proposed method.

POINTED REPRESENTATIONS OFINFINITE DIMENSIONAL LIE ALGEBRAS

A contravaried bilinear pairing X on every M(ρ) × M(ρθ) is determined and it is provedthat M(ρ)is irreducible if and only if K is left nondegellerate. It is also proved that every cyclicpointed module is a quotient of some Verma-like poillted module; moreover if it is irreduciblethen it is a quotieDt of the Vermarlike poiDted module by the left kernel of some bilinearpairing K. In case the mass fUnction is symmetric, there exists a bilinear form on M(ρ). It isproved that unitals pointed modules are integrable. In addition, a characterization of the massfunctions of Kac-Moody algebras is given, which is a generalization of the finite dimensionalLie algebras case.