Quasi-Exactly Solvable Differential Models: A Canonical Polynomials Approach

A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.

A New Class of Exactly Solvable Models within the Schrodinger Equation with Position Dependent Mass

The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.

One New Method to Obtain the Correlation Length of Solvable Models

We propose a new method to obtain the correlation length of gapped XXZ spin 1/2 antiferromagnetic chains. Following the relativistic quantum field theory in space-time dimensions, we use the exact dispersion of massive spinon to calculate the correlation length for XXZ spin 1/2 chain. We conjecture that the correlation length for other 1D lattice models can be obtained in the same way. Relation between dispersion and the oscillated correlation of gapped incommensurate lattice models is also discussed.

A Solvable Model in Two—Dimensional Gravity Coupled to a Nonlinear Matter Field

The two-dimensional gravity model with a coupling constant and a vanishing cosmological constant coupled to a nonlinear matter field is investigated. We found that the classical equations of motion are exactly solvable and the static solutions of the induced metric and scalar curvature can be obtained analytically. These solutions may be used to describe the naked singularity at the origin.

A New Quasi-Exactly Solvable Problem and Its Connection with an Anharmonic Oscillator

The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods.Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KStransformation.

Quasi-Exactly Solvable Time-Dependent Hamiltonians

A generalized method which helps to find a time-dependent Schr&#214dinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.

A SOLVABLE ONE-DIMENSIONAL ALLOCATION DECISION MODEL OF RENEWABLE RESOURCES

In the paper, the determinate atlecation decision model and the probabilistic allocation decision model of a kind of renewable resource are separatly studied by means of dynamic programming, and the optimal allocation policy is given under some special conditions.

On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

SOLVABILITY RESULTS OF A CONDITIONAL INPUT-OUTPUT EQUATION BASED ON A TYPE OF NONLINEAR LEONTIEF MODEL

A class of nonlinear and continuous type Leontief model and its corresponding conditional input-output equation are introduced, and two basic problems under the so called positive or negative boundary assumption are presented. By approaches of nonlinear analysis some solvability results of this equation and continuous perturbation properties of the relative solution sets are obtained, and some economic significance are illustrated by the remark.

Free dendritic growth model based on nonisothermal interface and microscopic solvability theory

Considering both the effect of nonisothermal nature of the solid/liquid interface and the microscopic solvability theory (MicST), a further improved version of free dendritic growth model for pure materials was proposed. Model comparison indicates that there is a higher temperature at the tip of dendrite predicted by the present model compared with the corresponding model with the isothermal solid/liquid interface assumption. This is attributed to the sidewise thermal diffusion, i.e. the gradient of temperature along the nonisothermal interface. Furthermore, it is indicated that the distinction between the stability criteria from MicST and marginal stability theory (MarST) is more significant with the increase of bath undercoolings. Model test also indicates that the present model can give an agreement with the available experimental data. It is finally concluded that the nonisothermal nature of the solid/liquid interface and the stability criterion from MicST should be taken into account in modeling free dendritic growth.

Effects of electron correlation on superconductivity in the Hatsugai-Kohmoto model

Understanding how electrons form pairs in the presence of strong electron correlations demands going beyond the BCS paradigm.We study a correlated superconducting model where the correlation effects are accounted for by a U term local in momentum space.The electron correlation is treated exactly while the electron pairing is treated approximately using the mean-field theory.The self-consistent equation for the pair potential is derived and solved.Somewhat contrary to expectation,a weak attractive U comparable to the pair potential can destroy the superconductivity,whereas for weak to intermediate repulsive U,the pair potential can be enhanced.The fidelity of the mean-field ground state is calculated to describe the strength of the elelectron correlation.We show that the pair potential is not equal to the single-electron superconducting gap for the strongly correlated superconductors,in contrast to the uncorrelated BCS limit.

THE DISLOC ATION EQU ATIONS OF A SIMPLE CUBIC CRYSTAL IN THE ISOTROPIC APPROXIMATION-A SOLVABLE MODEL

The dislocation equations of a simple cubic lattice have been obtained by using Green＇s function method based on the discrete lattice theory with the coefficients of the secondorder differential terms and the integral terms have been given explicitly in advance. The simple cubic lattice we have discussed is a solvable model, which is obtained according to the lattice statics and the symmetry principle and can verify and validate the dislocation lattice theory. It can present unified dislocation equations which are suitable for most of metals with arbitral lattice structures. Through comparing the results of the present solvable model with the dislocation lattice theory, it can be seen that, the coefficients of integral terms of the edge and screw components we obtain are in accordance with the results of the dislocation lattice theory, however, the coefficient of the second-order differential term of the screw component is not in agreement with the result of the dislocation lattice theory. This is mainly caused by the reduced dynamical matrix of the surface term, which is the essence to obtain the dislocation equation. According to the simple cubic solvable model, not only the straight dislocations but also the curved dislocations, such as the kink, can be investigated further.

The Quasi-Exactly Solvable Problems in Relativistic Quantum Mechanics

We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein–Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.

静态电压稳定风险评估

考虑系统控制设备的运行情况和静态电压失稳校正控制措施,以切负荷最小为目标函数,建立判断系统潮流可解性的全二次优化模型,并结合模态分析法来识别出系统失去静态电压稳定的状态,从而提出判断电压稳定性的新方法。基于所提方法,结合蒙特卡罗模拟法对系统状态进行随机抽样,从而对系统的静态电压稳定性进行风险评估,可以计算出系统静态电压失稳概率、维持电压稳定需要的切负荷量期望值以及切负荷量与系统静态电压失稳概率之间的关系,为电压稳定风险评估提出一条有效的途径。IEEE14节点系统的仿真结果验证了所提模型和方法的有效性。

9-羟基苯嵌萘酮的基态能级劈裂的理论研究

当γ,μ满足一定条件时 ,六次函数 V( R,γ) =C[R6 +2γR4 +(γ2 +μ) R2 ]的图像是双势阱 ,可用来描述质子转移 ,且其相应的 Schrodinger方程有解析解 .用此势函数结合一维双势阱模型及变分法 ,并用 B3 P86密度泛函方法在 6-3 1 1 G水平下计算了该分子的平衡态和过渡态的构型及能量 ,研究了 9-羟基苯嵌萘酮的基态能级劈裂 ,与其它理论值相比 ,所得计算结果 [ΔH( 0 ) =86.1 3 cm- 1

一个内生人口生育率可解的C-K模型

得到一个具有可解内生人口出生率的Ｃａｓｓ－Ｋｏｏｐｍａｎｓ经济增长模型．给出此模型在不同劳动和资本的结合方式下存在唯一最优增长轨道和稳态解的条件．最后讨论存在多重最优增长轨道和多重稳态解的可能性态和文中主要结果的经济意义．

非均衡模型的可解性研究

本文研究了多市场非均衡模型（ＭＤＥＭ）的可解性问题，给出微观ＭＤＥＭ可解性的一个充要条件；推广了Ｉｔｏ［１］的充分条件；对具有Ｂｕｒｋｅｔｔ双曲线聚合［２］和Ｌａｍｂｅｒｔ指数聚合［３］两类非线性形式的宏观ＭＤＥＭ，分别给出其可解性的一个充分条件，并探讨ＭＤＥＭ中一类线性溢出效应的微观基础，Ｉｔｏ［１］所分析的双市场情形是其特例．

离散空间上单目标3维e容错搜索模型探析

针对离散空间上单目标3维e容错搜索模型中的end状态s,从理论上给出了提问者能够搜索成功的最小提问次数是其特征ch(s)。

超细长弹性杆的动力学模型及动力学方程解的存在性

在kirchhoff弹性杆理论框架下,考虑可以扭转、弯曲、剪切、拉伸并受到内力作用的弹性杆模型.在截面主轴坐标系下,建立了利用剪切拉伸向量描述的运动弹性杆的动力学模型;根据Doliwa和Santin提出的曲线运动方程的可解性理论,利用Lax方程可解性,分析了该动力学模型解的存在性,为对这一类动力学方程进行数值求解和数值模拟提供了理论依据.

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.