51,903 research outputs found
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
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