There having been no monograph on Lyapunov

characteristic exponents to study the stability of ordinary differential equations since the 1970s, Izobov provides a brief overview of the technique's current status.

For the presentation of the Lyapunov lemma on characteristic exponents and for its deeper understanding, we will use two integral inequalities.

In doing so, the characteristic exponents will play a similar role to the roots of the characteristic equation for the system of differential equations with constant coefficients.

If all characteristic exponents are finite and if there exists the index p, 1 [less than or equal to] p [less than or equal to] n, such that

Now we can carry out a closer investigation of the set of characteristic exponents of the solution to the linear system described by the equation (1)with a bounded matrix (2).

0], +[infinity]> and having mutually different characteristic exponents are linearly independent.

The results in Table 3 show that the estimated characteristic exponents of all [p.

The first parameter, [alpha] [member of] (0, 2], is the characteristic exponent that accounts for the relative importance of the tails.

8) Table 3 displays the estimated parameters of the characteristic exponent, [alpha], and the skewness parameter, [beta], of the stable distribution, which are the most important parameters that direct the data to a specific distribution in the stable set.

4) Fama (1963) suggests, using the fact that a stable distribution is invariant under addition, that the distribution of sums of the stable distribution is also a stable distribution with the same values of [alpha], which is the most important parameter in the stable distribution and is called the characteristic exponent.

058 Notes: [alpha] is the characteristic exponent and [beta] is the skewness parameter of the stable distribution.

The approach computes the local Denef- Loeser motivic zeta function of a quasi-ordinary power series of arbitrary dimension over an algebraically closed field of characteristic zero from its

characteristic exponents without using embedded resolution of singularities.