A Linear System Arising from a Polynomial Problem and Its Applications

A linear system arising from a polynomial problem in the approximation theory is studied, and the necessary and sufficient conditions for existence and uniqueness of its solutions are presented. Together with a class of determinant identities, the resulting theory is used to determine the unique solution to the polynomial problem. Some homogeneous polynomial identities as well as results on the structure of related polynomial ideals are just by-products.     

Analysis of a Degenerate Obstacle Problem on an Unbounded Set Arising in the Environment

We study a class of optimization dynamics problems related to investment under uncertainty. The general model problem is reformulated in terms of an obstacle problem associated to a second-order elliptic operator which is not in divergence form. The spatial domain is unbounded and no boundary conditions are a priori specified. By using the special structure of the differential operator and the spatial domain, and some approximating arguments, we show the existence and uniqueness of a solution of the problem. We also study the regularity of the solution and give some estimates on the location of the coincidence set.     

Analysis of a Degenerate Obstacle Problem on an Unbounded Set Arising in the Environment

We study a class of optimization dynamics problems related to investment under uncertainty. The general model problem is reformulated in terms of an obstacle problem associated to a second-order elliptic operator which is not in divergence form. The spatial domain is unbounded and no boundary conditions are a priori specified. By using the special structure of the differential operator and the spatial domain, and some approximating arguments, we show the existence and uniqueness of a solution of the problem. We also study the regularity of the solution and give some estimates on the location of the coincidence set.     

Spectral Analysis of an Operator Arising in Fluid Dynamics

Eigenmode solutions are very important in stability analysis of dynamical systems. The set of eigenvalues of a non-self-adjoint differential operator originated from the linearization of some Cauchy problem is investigated. It is shown that the eigenvalues are purely imaginary, and that they are related to the eigenvalues of Heun's differential equation. These two results are used to derive the asymptotic behavior of the eigenvalues and to compute them numerically.     

对一道多项式问题的解析

多项式是高等代数的重要内容之一.本文通过对一道多项式问题的解析来阐述处理这类多项式问题的基本方法和技巧.特别地,借助中国剩余定理可以更清楚地看到该问题的本质.     

Stability Analysis of the Solution to a System of Nonlinear Integral Equations Arising in a Logistic Dynamics Model

Doklady Mathematics - In this paper, we analyze a system of nonlinear integral equations resulting from the three-parameter closure of the third spatial moments in the logistic dynamics model of U....     

Positive Solution of a Nonlinear Parabolic System Arising in Grain Drying

Coupled system of nonlinear parabolic equations for grain drying is proposed and the existence and uniqueness theorem of classical solutions is proved by using the upper and lower solutions technique. The long-time behaviour of the solution is also investigated.     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

Globally Smooth Solutions to an Inhomogeneous Quasilinear Hyperbolic System Arising in Chemical Engineering

In this paper we have obtained the existence of globally smooth solutions to an inhomogeneous nonstrictly hyperbolic system u_t - (v(1 - u))_x = 0, v_t + (\frac{1}{2}v² - c_0u)_x = f(u,v) by employing the characteristic method and the fixedpoint theorem in Banach spaces.     

Bayesian Analysis of Geostatistical Models With an Auxiliary Lattice

The Gaussian geostatistical model has been widely used for modeling spatial data. However, this model suffers from a severe difficulty in computation: it requires users to invert a large covariance matrix. This is infeasible when the number of observations is large. In this article, we propose an auxiliary lattice-based approach for tackling this difficulty. By introducing an auxiliary lattice to the space of observations and defining a Gaussian Markov random field on the auxiliary lattice, our model completely avoids the requirement of matrix inversion. It is remarkable that the computational complexity of our method is only O(n), where n is the number of observations. Hence, our method can be applied to very large datasets with reasonable computational (CPU) times. The numerical results indicate that our model can approximate Gaussian random fields very well in terms of predictions, even for those with long correlation lengths. For real data examples, our model can generally outperform conventional Gaussian random field models in both prediction errors and CPU times. Supplemental materials for the article are available online.     

Spectral Analysis of a Transport Operator Arising in Growing Cell Populations

The present paper is concerned with the spectral analysis of a transport-like operator derived from a model introduced by Rotenberg describing the growth of a cell population. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. The biological boundaries of μ = 0 and μ = a (a > 0) are fixed and tightly coupled through mitosis. At mitosis daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. We first discuss in detail the spectrum of the streaming operator for smooth and partly smooth boundary conditions. Next, we discuss the existence and nonexistence of eigenvalues of the transport operator in the half plane {λ ∈ ℂ : Reλ > where denotes the spectral bound of the streaming operator. In particular, the strict monotonicity of the leading eigenvalue (when it exists) of the transport operator with respect to different parameters of the equation is also considered. We close the paper by describing in detail the various essential spectra of the transport operator for wide classes of collision and boundary operators.     

一类退化半导体方程弱解存在性的研究

文章研究了当■(s)=sm(m>1),b(s)=s2和初值为u0,v0∈L 2(Ω)时一类退化半导体方程弱解的存在性．文章首先将原问题正则化,然后对正则化问题在L 2(Ω)空间上做出了有界估计,最后利用收敛性得到了问题的结论．     

一类六角系统图的特征多项式及应用

Let Ln be the hexagonal chain graph,Fnbe the hexacyclic system graph and Mn be the M¨obius hexacyclic system graph. Derflinger and Sofer gave the spectra of Ln and Fn by using group theoretical method. Later, Gutman gave the spectra of them using a polynomial result due to Godsil and McKay. In this paper, we give a simple and direct method to determine the characteristic polynomial and spectra of Fn and Ln. By the method, we give the characteristic polynomial and spectrum of Mn that is new. Additionally, the exact values of total π-electron energy and the nullities of Ln, Fn and Mn are obtained, and the bounds for the energy of Ln and Mn are also considered.     

Computation of the Characteristic Polynomial of an Endomorphism of a Free Module

We present two methods of computing the characteristic polynomial of an endomorphism of a free module over a commutative domain. The methods require O(n ³) and O(n ^log7) ring operations, respectively. Bibliography: 6 titles.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

具不变集的平面多项式系统

本文证明了以y＝xm为不变集的平面n次多项式系统（m＞n）不会有极限环，但可以存在奇闭轨．