This appears to be much more natural, since only the invertibility
of S is needed to get a time scale symplectic system.
Based on the definition of invertibility
, which has been described earlier, the result demonstrates clearly that the S-matrix based watermark embedding is invertible since the attacker was able to extract his faked watermark [W.
Characterization of W-maps in terms of Invertibility
, Bulletin of Pure and Applied Sciences.
measure preserving systems of differential equations or invertible maps) the conditional entropy is fixed at the value with which the system is prepared   and , but that the addition of noise can reverse this invertibility
property and lead to an evolution of the conditional entropy to a maximum value of zero.
Other concepts that appeared first in map E and remained unchanged in map F are those detailing invertibility
of matrices namely: det A, det=0, det [not equal to] 0, not invertible, invertible and augmented matrix.
Conditions for invertibility
and results on the accuracy of this VAR approximation can be found in Fernandez-Villaverde, Rubio-Ramirez, and Sargent (2004).
Under standard invertibility
conditions the systems have the following moving average representation:
However, a model with a moving average coefficient of approximately unity may be statistically uninterpretable due to its violation of an invertibility
condition (Chatfield 1989).
An advantage of the invertibility
exercise is that it allows access to, and relatively transparent applications of, a number of theorems in applied mathematics.
POLSYS_PLP uses numerical linear algebra with random real matrices to determine the generic invertibility
Amoroso and Patt proved that there is an effective procedure to determine invertibility
of 1-d CA, based on the local rule [Amoroso and Patt 1972].
trade-offs between the value of the long-memory parameter and those of the ARMA parameters as well as possible stationarity and invertibility
problems with the AR and MA polynomials, respectively).