Narrow Results

We construct a singular oscillator Hamiltonian with a position-dependent effective mass. We find that an su(1, 1) algebra is the hidden symmetry of this quantum system and the isospectral potentials V(x) depend on the different choices of the m(x). The complete solutions are also presented by using this Lie algebra.

By a simple algebraic approach we study the exact solution to the Dirac equation with scalar and vector trigonometric Scarf potentials in the case of spin symmetry. The transcendental energy equation and spinor wave functions are presented. It is found that there exist only positive energy bound states in the case of spin symmetry. Also, the energy eigenvalue approaches a constant when the potential parameter α goes to zero. The equally scalar and vector case is studied briefly.

We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short-periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Šilnikov bifurcation which clarifies the presence of nonintegrability phenomena qualitatively and quantitatively.

Students often experience problems in solving time-value-of-money problems — primarily because they do not know which interest rate to use from among the nominal, effective, and periodic rates. In this paper we show the relationships among different interest rates and clarify the use of these rates in the time-value-of money problems. In particular, we show how to calculate the periodic rate, given the nominal rate, because students must use the periodic rate when they use either the “formula” or “financial cal culator” method to compute the interest rate.

Students often experience problems in solving time-value-of-money problems — primarily because they do not know which interest rate to use from among the nominal, effective, and periodic rates. In this paper we show the relationships among different interest rates and clarify the use of these rates in the time-value-of money problems. In particular, we show how to calculate the periodic rate, given the nominal rate, because students must use the periodic rate when they use either the “formula” or “financial calculator” method to compute the interest rate.