Starting algorithms for low stage order RKN methods

When second order differential equations are solved with Runge-Kutta-Nyström methods, the computational effort is dominated by the cost of solving the nonlinear system. That is why it is important to have good starting values to begin the iterations. In this paper we consider a type of starting algorithms without additional computational cost. We study the general order conditions and the maximum order achieved when the Runge-Kutta-Nyström method satisfies some simplifying assumptions.     

On a Reliable Solution of a Volterra Integral Equation in a Hilbert Space

We consider a class of Volterra-type integral equations in a Hilbert space. The operators of the equation considered appear as time-dependent functions with values in the space of linear continuous operators mapping the Hilbert space into its dual. We are looking for maximal values of cost functionals with respect to the admissible set of operators. The existence of a solution in the continuous and the discretized form is verified. The convergence analysis is performed. The results are applied to a quasistationary problem for an anisotropic viscoelastic body made of a long memory material.     

Hilbert空间的正交分解及其应用

基于Riesz-表示算子,给出了实Hilbert内积空间按某种连续双线性泛函的正交分解的刻划,应用于鞍点变分问题,获得了解的分离及其强制型于问题.     

Implicit kinetic schemes for scalar conservation laws

Based on kinetic formulation for scalar conservation laws, we present implicit kinetic schemes. For time stepping these schemes require resolution of linear systems of algebraic equations. The scheme is conservative at steady states. We prove that if time marching procedure converges to some steady state solution, then the implicit kinetic scheme converges to some entropy steady state solution. We give sufficient condition of the convergence of time marching procedure. For scalar conservation laws with a stiff source term we construct a stiff numerical scheme with discontinuous artificial viscosity coefficients that ensure the scheme to be equilibrium conserving. We couple the developed implicit approach with the stiff space discretization, thus providing improved stability and equilibrium conservation property in the resulting scheme. Numerical results demonstrate high computational capabilities (stability for large CFL numbers, fast convergence, accuracy) of the developed implicit approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 26–43, 2002     

Minimizing flow time on a constant number of machines with preemption

We consider offline algorithms for minimizing the total flow time on O(1) machines where jobs can be preempted arbitrarily but migrations are disallowed. Our main result is a quasi-polynomial time approximation scheme for minimizing the total flow time. We also consider more general settings and give some hardness results.     

两个重要不等式

本文引入了 P[a](x)类的权函数 ,定义了一类新型卷积 ,推广了 Hilbert不等式 ,并证明了关于卷积的一个新型精确不等式     

Symplectic Methods for the Numerical Solution of the Radial Shrödinger Equation

In this paper new symplectic-schemes for the numerical solution of the radial Shrödinger equation are proposed. In particular, symplectic integrators for Hamiltonian systems have been developed. Based on this approach, second- and third-order methods are proposed. These methods are more accurate than the existing ones. We compare these methods not only with the existing symplectic methods, but also with a classical Runge–Kutta–Nyström method.     

When modified Gram-Schmidt generates a well-conditioned set of vectors

In this paper, we show why the modified Gram–Schmidt algorithmgenerates a well-conditioned set of vectors. This result holdsunder the assumption that the initial matrix is not ‘tooill-conditioned’ in a way that is quantified. As a consequencewe show that if two iterations of the algorithm are performed,the resulting algorithm produces a matrix whose columns areorthogonal up to machine precision. Finally, we illustrate througha numerical experiment the sharpness of our result.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

Some mathematical aspects of the political business cycle

Under the hypothesis of a myopic electorate, vote-loss-minimizing behavior by the party in power, subject to a dynamic inflation-unemployment relation, is shown to generate an attractive, stable electoral policy cycle. The model presented is derived, with some improvements, from the analogous models of MacRae and Nordhaus. Furthermore, an attempt is made to specify the mathematical aspects of the problem by the Poincaré mapping.This work was realized within the activities of CNR, Gruppo Nazionale Analisi Funzionale (GNAFA). The author wishes to thank Professor M. D. Intriligator for his many comments, suggestions, and critique.