单位圆内代数体函数的Borel点和Nevanlinna点

本文对单位圆内的代数体函数 w(z)定义了 Borel 点和 Nevanlinna 点,证明了 Nevanlinna 点的存在性,并在 w(z)的级为有穷时,亦证明了 Borel 点的存在性。     

一道例题的引申与启发

通过对一道关于递推数列求极限例题的引申和探索,说明教师可以启发和引导学生进行变换、推广和类比,使其有所新发现．     

Shallow water waves over an uneven bottom and an inhomogeneous KP equation

Three-dimensional shallow water waves over an uneven bottom are considered. The depth is assumed to be slow in variation. As a model, an inhomogeneous Kadomtsev-Petviashvili equation is presented. Some reductions of this equation are used to describe deformation of a line soliton due to the depth change. The model equation is valid for a wide class of two-dimensional nonlinear waves in inhomogeneous systems.     

Formulating an objective for an economy

This paper is dedicated to the memory of E.M.L. Beale. Not a small part of Martin Beale's success in developing and solving large-scale mathematical programs is attributed to the care he took in properly formulating his models.Our presentation concerns our efforts to also properly formulate a model. Our model is an economic model used for technology assessment. In order for it to be useful, it is important that the dual variables represent as realistically as possible real world prices. This required us to formulate the model as a time-staged economic equilibrium model. Our main result is a proof that an equilibrium formulation using expected aggregate demand can under certain conditions be replaced by one in which the economy is driven by an aggregate utility or objective function, one that promotes economic growth subject to physical flow constraints. We show that such an objective function always exists except for populations consisting of significantly large classes of people whose consumption patterns differ radically one from another. Assuming that the latter is not the case, this equivalent formulation means that mathematical programming software can be applied to efficiently solve the model. This paper summarizes an extensive paper entitled Deriving a Utilty Function for the U.S. Economy [3]. The main theorems are stated without proof.The authors wish to thank Kenneth Arrow, Gerard Debreu, Robert Dorfman, Dale Jorgenson and Lawrence J. Lau for their helpful comments.Research of this report was partially supported by the National Science Foundation Grants DMS-8420623, SES-8518662 and ECS-8617905; U.S. Department of Energy Grant DE-FG03-87ER25028; Office of Naval Research Contract N0004-85-K-0343, Electric Power Research Institute Contract RP 5006-01, and the Center for Economic Policy Research at Stanford University.     

Optimal investment and risk control policies for an insurer in an incomplete market

In this paper, we apply the martingale approach to investigate the optimal investment and risk control problem for an insurer in an incomplete market. The claim risk of per policy is characterized by a compound Poisson process with drift, and the insurer can be invested in multiple risky assets whose price processes are described by the geometric Brownian motions model. By ‘complete’ the incomplete market, closed-form solutions to the problems of mean–variance criterion and expected exponential utility maximization are obtained. Moreover, numerical simulations are presented to illustrate the results with the basic parameters.     

An L-Point Characterization of Normality and Normalizer of an L-Subgroup of an L-Group

In this paper, we study the notion of normal $L$-subgroup of an $L$-group and provide its characterization by an $L$-point. We also provide a construction of the normalizer of an $L$-subgroup of a given $L$-group by using $L$-points. Moreover, we also discuss the product, homomorphic images and homomorphic preimages of normalizers.     

Some Simple Remarks on an Autoregressive Scheme and an Implied Problem

Nonminimum phase autoregressive schemes are considered in the context of the problem of determining a stationary process that satisfies a possibly nonlinear autoregressive system of equations. It's noted that in most solutions there is an implicit assumption that the independent random variables generating the process are independent of the past of the process. This is not true of the non-minimum phase schemes. A simple scheme that has a characteristic polynomial with roots inside and outside the unit disc in the complex plane is discussed.     

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.     

Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.     

The equilibrium of an elastic space weakened by two spherical cavities and an external circular crack

Using the relationship between the basic solutions of Laplace's equation in toroidal and spherical coordinates, the Fourier method is employed to solve the problem of the equilibrium of an elastic space weakened by two spherical cavities and an external circular crack. The proposed approach leads to an infinite system of linear algebraic equations of the second kind with exponentially decaying matrix coefficients. A small-parameter expansion is used to obtain an asymptotic formula for the normal stress intensity factor.     

On an extension problem,generalized fourier analysis,and an entropy formula

Let h(t) be an n × n matrix valued function on the interval |t| with summable entries. Let denote the Fourier transform of h and let e denote the n × n identity matrix. Necessary and sufficient conditions for the existence of an extension u of h to the full line such that e-û admits either a left or a right canonical factorization and the inverse transform of (e-û)^–1-e vanishes for |t| are presented and discussed. The connections between these extensions and a generalized Fourier transform are then explored in detail with the help of the theory of triangular factorization. It is then shown that if an allied finite Wiener-Hopf operator based on h is positive, then h admits exactly one extension of the type alluded to above. This extension is then characterized in terms of an entropy integral.     

Minimal circumscribed Hermitian ellipsoid of Hartogs domains and an application to an extremal problem

In this paper we construct circumscribed Hermitian ellipsoids of Hartogs domains of least volume and as an application, we obtain the Carath′eodory extremal mappings between the Hartogs domains and the unit ball, and also give an explicit formula for calculating the extremal values.     

Mathematics analysis and chaos in an ecological model with an impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model with an impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest-level prey and mid-level predator eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems.     

Intensive quantities in an economy and conjugate variables

In an economy, the flows of goods, resources, and money are extensive (additive) variables, while the prices, interest, and exchange rates serve as intensive variables. Using the simplest economic planning problem as an example, we show that these intensive variables arise naturally as conjugate variables to balance constraints under the decomposition of the planning problem into the problem of the dynamical (intertemporal) equilibrium of an economy. It is also shown that linear homogeneity symmetry, which is a natural feature of an economy, generates, by Noether’s theorem, an equity capital equation.     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

平面P波在弹性介质和非饱和多孔弹性介质分界面上的传播

使用线性粘滞的多孔弹性介质模型,解决在弹性介质和非饱和多孔弹性介质分界面上平面P波的反射与透射问题,这里的非饱和多孔介质中固体骨架被两种相互耦合的流体（液体和气体）所充满．通过势函数的方法得到了振幅反射系数与振幅透射系数．然后推导得到入射波与反射波、透射波之间能量转换情况．研究发现:用振幅比和能量比所表示的反射系数与透射系数是与入射角度、饱和度、入射频率以及上下层介质的弹性常数有关的方程式．数值计算通过图形的形式表达出来,而且入射角度、频率及饱和度对振幅和能量的反射与透射系数的影响分别进行了讨论．证明了在整个波的传播过程中分界处并没有发生能量的耗散．     

Inequalities for a Simplex and an Interior Point

We establish some inequalities for the inradius, circumradius and distances between an interior point and facets of an n-simplex, and prove a recursion inequality for the radii of the circumscribed spheres of an n-simplex and its facets.     

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 34–57, April, 2009.     

On the existence and computation of an equilibrium in an economy with constant returns to scale production

In this paper we consider the problem of finding an equilibrium in an economy with non-linear constant returns to scale production activities. To find an equilibrium we propose an adjustment process in which the prices of the commodities and the activity levels of production adjust simultaneously. The process starts at a price vector at which each production activity has non-positive profit. We show that the process follows a path which connects the starting point with an equilibrium of the economy. From this it follows that the existence of a price vector at which each production activity has non-positive profit implies the existence of an equilibrium. The equilibrium can be computed by using a simplicial algorithm or by solving a sequence of Linear Variational Inequality Problems.This research is part of the VF-program Competition and Cooperation. The authors are very grateful to Dolf Talman and two unknown referees for their valuable comments and suggestions.