The time scale logarithm

The Natural Logarithm on Time Scales

We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. 11 (2005), no. 15, 1305–1306].

The Logarithm Function with a Fuzzy Time Series

The fuzzy time series has recently received increasing attention because of its capability in dealing with vague and incomplete data. This article presents an improved fuzzy time series model; we show that our method is as complete as the original definition but with higher reliability (Chou and Lee). Experimental results using the University of Alabama’s enrollment data (adapted by Song and Ch...

Laws of the iterated logarithm for α-time Brownian motion

We introduce a class of iterated processes called α-time Brownian motion for 0 < α ≤ 2. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric α-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in [14] for iterated Brownian motion. When α = 1 it takes the following form l...

Problem 3 : The Discrete Logarithm

Lie algebra generated by logarithm of differentiation and logarithm

Let log ( d dx ) be the generator of the1-parameter group { d dxa |a ∈ R} of fractional order differentiations acting on the space of operators of Mikusinski ([5]). The Lie algebra glog generated by log ( d dx ) and log x is a deformation and can be regarded as the logarithm of Heisenberg Lie algebra. We show glog is isomorphic to the Lie algebra generated by d ds log(Γ(1 + s)) and d ds . Hence...