In this paper, we study the stabilization problem of a flexible cable with two rigid loads which is described by two kinds of hyperbolic equations. From the engineering point of view, the model can be regarded as a distributed parameter overhead crane which conveys two loads simultaneously. After deriving a control law which does not increase an energy defined for the model, we prove the asymptotic stability of the closed-loop system, using the LaSalle's invariance principle. Finally, we show that the control law works effectively through a numerical simulation.

A new scheme with a shift of origin for computing singular values $\sigma_{k}$ is presented. A shift $\theta$ is introduced into the recurrence relation defined by the discrete integrable Lotka--Volterra system with variable step-size. A suitable shift strategy is given so that the singular value computation becomes numerically stable. It is proved that variables in the new scheme converge to $\sigma_{k}^{2}-\sum\theta^{2}$. A comparison of the zero-shift and the nonzero-shift routines is drawn. With respect to both the computational time and the numerical accuracy, it is shown that the nonzero-shift routine is more accurate and faster than a credible LAPACK routine for singular values at least in four different types of test matrices.

High magnetic force field plays an important role in the wide applications of magnetic processing such as magnetic separation of ferromagnetic materials and magnetic levitation of diamagnetic materials. Therefore, the research on magnetic force field enhancement has been becoming a popular subject recent years. Many experimental methods have been designed and applied to enhance magnetic force field, however those methods took higher costs and longer time to set up real installations. A numerical optimization method was developed, which used only superconducting coils to calculate a magnetic force field and maximize it. In the method, the magnetic flux density $B$, the magnetic field gradient $\grad B$ and the magnetic force field $\grad (B^{2}/2)$, as objective functions respectively, were maximized with constraints of the total volumes of coils and the $B$-$J$ characteristics of superconductors. By comparing the numerical optimization results with theoretical values for single coil, we found that they are in good agreement. Also, in comparison with the results obtained by random models for double coils, those by optimal models were better. Moreover, the maximal magnetic force field was higher remarkably than that generated by the previous optimal model in NIMS.

This paper is concerned with an indefinite weight linear eigenvalue problem which is related with biological invasions of species. We investigate the minimization of the positive principal eigenvalue under the constraint that the weight is bounded by a positive and a negative constant and the total weight is a fixed negative constant. For an arbitrary domain, it is shown that every global minimizer must be of ``bang-bang'' type. When the domain is an interval, it is proved that there are exactly two global minimizers, for which the weight is positive at one end of the interval and is negative in the remainder. The biological implication is that a single favorable region at one end of the habitat provides the best opportunity for the species to survive, and also that the least fragmented habitat provides the best chance for the population to maintain its genetic variability.

This paper is concerned with semilinear tow-point boundary value problems of the form $-(p(x)u')'+f(x,u)=0$, $a\leq x\leq b$, $\alpha_{0}u(a)-\alpha_{1}u'(a)=\alpha$, $\beta_{0}u(b)+\beta_{1}u'(b)=\beta$, $\alpha_{i}\geq 0$, $\beta_{i}\geq 0$, $i=0,1$, $\alpha_{0}+\alpha_{1} > 0$, $\beta_{0}+\beta_{1} > 0$, $\alpha_{0}+\beta_{0} > 0$. Under the assumption $\inf f_{u} > -\lambda_{1}$, where $\lambda_{1}$ is the smallest eigenvalue of $\mathrsfs{L}u=-(pu')'$ with the boundary conditions, unique existence theorems of solution for the continuous problem and a discretized system with not necessarily uniform nodes are given as well as error estimates. The results generalize three theorems of Lees for $u''=f(x,u)$, $0\leq x\leq 1$, $u(0)=\alpha$, $u(1)=\beta$.

We have made the implementation of symbolic computation programs which can derive identity relations for arbitrarily given mathematical expressions. Simulations of relatively easy several concrete examples have been shown to run within practical speeds.