On the exact discretization of the classical harmonic oscillator equation

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.Comment: 29 page

Gender Differences in Risk Perception: Broadening the Contexts

The author surveys literature on the effect of gender on risk perception

Some implications of a new definition of the exponential function on time scales

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Pad\'e approximants, we obtain Pad\'e-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.Comment: 12 pages. Presented at 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications; Dresden, 25-28.05.201

New definitions of exponential, hyperbolic and trigonometric functions on time scales

We propose two new definitions of the exponential function on time scales. The first definition is based on the Cayley transformation while the second one is a natural extension of exact discretizations. Our eponential functions map the imaginary axis into the unit circle. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way. The resulting functions preserve most of the qualitative properties of the corresponding continuous functions. In particular, Pythagorean trigonometric identities hold exactly on any time scale. Dynamic equations satisfied by Cayley-motivated functions have a natural similarity to the corresponding diferential equations. The exact discretization is less convenient as far as dynamic equations and differentiation is concerned.Comment: 27 page