Asymptotics,Stability, and Region of Attraction of Periodic Solution to a Singularly Perturbed Parabolic Problem with Double Root of a Degenerate Equation

For a singularly perturbed parabolic problem with Dirichlet boundary conditions, the asymptotic decomposition of a solution periodic in time and with boundary layers near the ends of the segment where the degenerate equation has a double root is constructed and substantiated. The construction algorithm for the asymptotics and the behavior of the solution in the boundary layers turn out to differ significantly as compared to the case of a simple root of a degenerate equation. The stability of the periodic solution and its region of attraction are also studied.     

方程的求根公式和根的展开式及其算法

本文获得了方程的求根公式及其根的级数展开式，并得到了求根的算法，同时，建立了求根问题的一个新方法——函数零点和方程根公式表示法。     

一元n次多项式根的展开公式及其求根算法

本文获得了一元n次复系数多项式根的展开公式，给出了求出方程的任意精确根的一个新的算法。利用该算法，可以用求根公式得到任意精度的初始值，用一个公式可以计算出全部的根。     

Kernel-Splitting Technique for Enclosing the Solution of Fredholm Equations of the First Kind

We derive a numerical method for solving linear Fredholm integral equations of the first kind. Based on series expansion techniques, the kernel of the corresponding integral equation is splitted into a finite rank degenerate part and an infinite dimensional, normwise small remainder. By enclosing the remainder term, the original problem, is transformed into a degenerate set-valued problem. For this problem, we derive a numerical method that provides a rigorous control of approximation and roundoff errors. We show that this approach provides a regularization scheme.     

Investigation of the dynamics of a double pendulum with following and conservative forces

For a double pendulum loaded by following and conservative forces, in the nonlinear statement, the problem of stability of its zero equilibrium position is solved for the following critical stability cases: a single zero root, a pair of purely imaginary roots, a single zero root and a pair of purely imaginary roots of the characteristic equation. All results of the analytical computations are illustrated by numerical simulation.     

Symmetry breaking and the twisted elastic ring

Symmetry breaking is considered in the context of a parameter-dependent two-point boundary value problem describing a twisted elastic ring that arises as a model of DNA minicircles. We explicitly determine, via a perturbation expansion, those representatives of a manifold of solutions to a symmetric BVP that persist when the symmetry is broken. Conditions generating a generic splitting are identified, and some degenerate cases are studied. In both the generic and degenerate cases, the results of the perturbation expansion are used to improve the efficiency of existing algorithms for computing symmetry-broken bifurcation diagrams.     

Rigid body mode and spurious mode in the dual boundary element formulation for the Laplace problems

In this paper, the general formulation for the static stiffness is analytically derived using the dual integral formulations. It is found that the same stiffness matrix is derived by using the integral equation no matter what the rigid body mode and the complementary solutions are superimposed in the fundamental solution. For the Laplace problem with a circular domain, the circulant was employed to derive the stiffness analytically in the discrete system. In deriving the static stiffness, the degenerate scale problem occurs when the singular influence matrix can not be inverted. The Fredholm alternative theorem and the SVD updating technique are employed to study the degenerate scale problem mathematically and numerically. The direct treatment in the matrix level is achieved to deal with the degenerate scale problems instead of using a modified fundamental solution. The addition of a rigid body term in the fundamental solution is found to shift the zero singular value for the singular matrix without disturbing the stiffness.     

Revisit of the Indirect Boundary Element Method: Necessary and Sufficient Formulation

Although the boundary element method (BEM) has been developed over forty years, the single-layer potential approach is incomplete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. The indirect boundary element method (IBEM) is revisited to examine the uniqueness of the solution by using the necessary and sufficient boundary integral equation (BIE). For the necessary and sufficient BIE, a free constant and an extra constraint are simultaneously introduced into the conventional IBEM. The reason why a free constant and an extra constraint are both required is clearly explained by using the degenerate kernel. In order to complete the range of the IBEM lacking a constant term in the case of a degenerate scale, we provide a complete base with a constant. On the other hand, the formulation of the IBEM does not contain a constant field in the degenerate kernel expansion for the exterior problem. To satisfy the bounded potential at infinity, the integration of boundary density is enforced to be zero. Besides, sources can be distributed on either the real boundary or the auxiliary (artificial) boundary in this IBEM. The enriched IBEM is not only free of the degenerate-scale problem for the interior problem but also satisfies the bounded potential at infinity for the exterior problem. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to illustrate the validity and the effectiveness of the necessary and sufficient BIE.     

GF(2~N)上线性化多项式根的存在性

给出了判断有限域上某元素为线性化多项式根的方法。在此基础上,将线性化多项式根的求解问题归结为有限域上简单方程的求解,使得可以方便地用计算机求解线性化多项式的根。     

A subdivision method for computing nearest gcd with certification

A new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The exclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smale’s α-theorems to certify the existence and unicity of a solution in a cell.Under the condition of simple roots for the distance minimization problem, we analyze the complexity of the algorithm in terms of a condition number, which is the inverse of the distance to the set of degenerate systems.We report on some experimentation on representative examples to illustrate the behavior of the algorithm.     

Robust and Efficient Ray Intersection of Implicit Surfaces

The generation of ray traced images of a variety of surfaces plays a central role in computer graphics. One of the main operations in ray tracing is the calculation of intersections between rays and surfaces. In case of implicitly given surfaces the intersection problem can be formulated as that of finding the smallest non-negative root of an equation in one variable. If the root finding is carried out by means of conventional numerical methods based on point sampling (such as bisection, regula-falsi or Newton) the resulting image can be wrong, e.g. when the surface is thin the ray may "miss" the surface, which may result in an image with background color spots on the surface. To obtain robust intersection detection, methods based either on Lipschitz constants for the function and its derivative or an interval inclusions for the function and its derivative have been suggested. In this paper robust methods are obtained with interval inclusions in a variant of Alefeld-Hansens globally convergent method for computing and bounding all the roots of a single equation. Alefeld-Hansens method has been modified so instead of searching for all roots, a recursive depth-first search is carried out to obtain the smallest non-negative root. When compared to other methods suggested, it is found that this variant of Alefeld-Hansens method is not only robust but also an efficient method for finding the ray intersections.     

Design of a single-input system for specified roots using output feedback

A method is given for choosing feedbacks from an output vector of smaller dimension than the state vector in such a way that a subset of the roots assume desired values. A sufficient condition for arbitrary placement of as many roots as there are feedbacks is proved. It is also proved that a root corresponding to a mode which is either uncontrollable or unobservable cannot be altered. It may be desired to place more roots than there are feedbacks, and it is shown that the method can be extended to give an approximate solution of this problem.     

Uplink power adjustment in wireless communication systems: a stochastic control analysis

This paper considers mobile to base station power control for lognormal fading channels in wireless communication systems within a centralized information stochastic optimal control framework. Under a bounded power rate of change constraint, the stochastic control problem and its associated Hamilton-Jacobi-Bellman (HJB) equation are analyzed by the viscosity solution method; then the degenerate HJB equation is perturbed to admit a classical solution and a suboptimal control law is designed based on the perturbed HJB equation. When a quadratic type cost is used without a bound constraint on the control, the value function is a classical solution to the degenerate HJB equation and the feedback control is affine in the system power. In addition, in this case we develop approximate, but highly scalable, solutions to the HJB equation in terms of a local polynomial expansion of the exact solution. When the channel parameters are not known a priori, one can obtain on-line estimates of the parameters and get adaptive versions of the control laws. In numerical experiments with both of the above cost functions, the following phenomenon is observed: whenever the users have different initial conditions, there is an initial convergence of the power levels to a common level and then subsequent approximately equal behavior which converges toward a stochastically varying optimum.     

Computation of simple and multiple roots of a nonlinear equation

Classical and generalized Newton’s methods as applied to the calculation of simple and multiple roots of a nonlinear equation are examined. For these two methods the algorithms are developed for determining the root’s order during calculations. For multiple roots the algorithm that accelerates convergence is developed. It also allows us to determine a root even to a high order with high accuracy. The details of all algorithms are examined carefully. Based on that, the program that surpasses well-known standard programs in accuracy and reliability is developed.     

一个数据膨胀率为1的概率公钥密码系统

在RSA公钥密码的基础上,采用时间戳和hash函数技术,并利用以Blum数为模的二次同余式中求平方根的不可计算性,设计了一个概率公钥密码系统,此密码系统的密码强度不低于RSA的密码强度和求以Blum数为模的二次同余式平方根的难度,加、解密的时间复杂度为O（k^3）,其中k为模数的长度,密码的数据膨胀率等于1,因此在数据膨胀率上,此概率公钥密码系统是最优的。     

基于实数多项式的根值最小范数算法

在根值最小范数算法中须对复数多项式求根,计算量较大。针对此问题,提出了一种基于实数多项式的根值最小范数算法,该算法适用于均匀线性阵列。首先通过保角变换将分布在复平面单位圆的变量映射到实数轴的[-1,1]范围内,其能够将算法中的复数多项式转换为实数多项式;其次对该实数多项式求根,并从中选出[-1,1]范围内的根值;最后将筛选出的根值代入信号频谱函数中,根据频谱函数的值选择出最优的波达方向估值。理论分析说明本文算法比根值最小范数算法的时间复杂度低;仿真实验表明,与根值最小范数相比,在信号和噪声不相关时,本文算法的均方根误差略小,在信号和噪声相关时,随着信噪比的增加,本文算法的均方根误差逐渐变小。     

指数函数多项式的实根分离算法

针对超越函数多项式的实根分离问题，提出了一种指数函数多项式的区间分离算法exRoot，将非多项式型实函数的实根分离问题转化为多项式正负性判定问题进而对其求解。首先，利用泰勒替换法构造目标函数的多项式区间套；然后，将指数函数的求根问题转化为多项式在区间内正负性的判定问题；最后，给出综合算法，并且试探性地应用于实特征值线性系统的可达性判定问题。所提算法在Maple中实现，输出的结果可读，且高效易行。区别于HSOLVER和数值计算方法fsolve，exRoot回避了直接讨论根的存在性问题，理论上具有终止性和完备性，且可达到任意精度，应用于最优化问题时可避免数值解带来的系统误差。     

Transfer of a Dynamic Object onto the Surface of an Ellipsoid

Using Pontryagin’s maximum principle, the problem of the quickest transfer of a multidimensional object onto the surface of an ellipsoid is reduced to solving a scalar algebraic equation. The concentration of the endpoints of optimal trajectories in the vicinity of the points forming the boundary in the case of a degenerate ellipsoid is demonstrated. An example in which the optimal control has a jump and the Bellman function has a discontinuity when the magnitude of the initial velocity vector undergoes a small change is constructed. It is also shown that the jump in the optimal control can occur without the discontinuity of the Bellman function.     

B样条曲面方向投影问题的几何计算方法

B样条曲面方向投影问题可以通过求解方程组的方法来解决.由于方程组所有根中往往只有一个或甚至没有根与待求解的最近点对应,因而绝大多数的求根计算量是不必要的.为此讨论了B样条曲面的方向投影问题,提出一种简单且高效稳定的几何计算方法.该方法充分利用了B样条函数的凸包性,同时结合B样条函数稳定可靠的分裂算法给出了相应的几何剪枝方法.与传统的求解非线性方程组的计算方法相比,文中方法可以剪除绝大部分非线性方程组对应的根,且不需要Newton迭代,可以应用于平面/B样条曲面间的求交测试问题及B样条曲面包围盒的计算问题.实例结果表明,该方法具有比传统的相关方法更高的计算效率和更好的稳定性.     

Root isolation of zero-dimensional polynomial systems with linear univariate representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the complex roots of the polynomial system are represented as linear combinations of the roots of several univariate polynomial equations. An algorithm is proposed to compute such a representation for a given zero-dimensional polynomial equation system based on Gröbner basis computation. The main advantage of this representation is that the precision of the roots of the system can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain roots of the polynomial system with a given precision. As a consequence, a root isolating algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.