Positive definiteness in the numerical solution of Riccati differential equations

Summary. In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization. Received April 21, 1993     

Runge-Kutta schemes that maintain invariant solutions for differential algebraic systems

Implicit Runge-Kutta (IRK) methods and projected IRK methods for the solution of semiexplicit index-2 systems of differential algebraic systems (DAEs) have been proposed by several authors. In this paper we prove that if a method satisfiesBA+A ^t B–bb ^t =0, it conserves quadratic invariants of DAEs.     

Spontaneous translation symmetry breakdown for quantum lattice systems

We study the set of equilibrium states for quantum lattice states in the presence of a translation symmetry of the model. We derive a characterization of the spontaneous breaking of this symmetry, i.e., the decomposition of an invariant equilibrium state into a mixture of noninvariant equilibrium states, in terms of the separability in mean energy of these states for a class of perturbed dynamics.     

A 0-stable linear multistep formulas of theα-type

The-type linear multistep formulas are a generalization of the Adams-type formulas. This paper is concerned with completely characterizing theA ₀-stability of thek-step, orderk -type formulas. Specifically, all such formulas of orders 4 or less are identified and it is shown that no-type formulas of order 5 or more exist. These theorems generalize some previous results.     

Effect of Zr concentration on the mechanical and thermodynamic properties of NbIr3 intermetallic compounds from theoretical estimations

ABSTRACT

The iridium is an important metal which has excellent resistance to corrosion at high temperature. L1₂ intermetallic compounds i.e. Ir₃Nb and Ir₃Zr, with similar lattice parameters are ideal for working at high temperature. They are fully soluble due to their low lattice misfit. A first-principle investigation into the effect of doping Zr with different concentrations on the electronic structure, mechanical and thermodynamic properties of NbIr₃ has been studied to prompt the development of novel high-temperature materials. Nine Zr_xNb_8?xIr₂₄ compounds are carefully considered. The results show that adding Zr into these compounds can strengthen their structural stability and ductility. Nevertheless, it reduces the elastic modulus and elastic stiffness. Simultaneously, with the increase of Zr content, the thermodynamic properties of these compounds decrease. It is also found that the changes of elastic modulus are mainly attributed to the variations of bonds in these compounds.     

利用同质异位素产额比提取对称能的近似方法

在修改的Fisher模型框架下, 利用同质异位素产额比和结合能方程中对称能系数与温度比率的关联, 可以提取对称能系数与温度的比值。 利用3种不同的近似方法得到了重离子核反应产生丰中子余核的对称能系数与温度比（a_sym/T）,并研究了相关物理量对a_sym/T的影响。结果表明, 库仑能对提取丰中子余核的a_sym/T影响较小,且参考核的选取对a_sym/T的提取也有一定的影响。 In the framework of the modified Fisher model, correlations between the symmetry energy coefficient in semi classical mass formula to temperature (a_sym/T) and the isobaric yield ratio in Heavy ion collisions are established. a_sym/Tof neutron rich fragments are extracted using these correlations. It is found that the Coulomb energy term has little effect on a_sym/T of fragments, while the reference isobars adopted have great influence on the extracted a sym/T of fragments.     

The Lagrangian and the Lie symmetries of charged particle motion in homogeneous electromagnetic field

In this paper, a constant of motion of charged particle motion in homogeneous electromagnetic field is derived from Newton's equations and the characteristics of partial differential equation, the related Lagrangian is also given by means of the obtained constant of motion. By discussing the Lie symmetry for this classical system, this paper obtains the general expression of the conserved quantity. It is shown that the conserved quantity is the same as the constant of motion in essence.     

Phase diagrams of a spin-1 Ising superlattice

The three-dimensional spin-1 Ising superlattice consisting of two different ferromagnetic materials with two different crystal fields _Δ1${\Delta }_1$ and _Δ2${\Delta }_2$ is considered in the mean field approximation. The phase diagrams are considered in the (t,_d2$t,d_2$) plane for different ranges of variation of _d1(t=T/J,_d1=_Δ1/J$d_1(t=T/J,d_1={\Delta }_1/J$, _d2=_Δ2/J$d_2={\Delta }_2/J$ are the reduced temperature and crystal fields respectively). The phase diagrams exhibit a variety of multicritical points and reentrant and double reentrant behaviours. They are found to depend qualitatively and/or quantitatively on the thicknesses of the materials in a supercell. This has direct consequences on the nature of the magnetic states of superlattices with different thicknesses.