The criterion follows from a detailed study of the shift invariant subspaces of two-variable analytic functions.
In order to obtain this result, we show that X is regular if its spectral density is factorizable by an outer two-variable analytic (in the sense of Helson-Lowdenslager) function.
Here we extend this study to the complex variable for analytic functions in compact disks.
As the problem described by Eq. (21) is complicated nonlinear question, it is difficult to obtain the optimal control variables by analytic method.
Using this array of variables an analytic model represented by the following diagram (Fig. 1) was used to plan analyses using hierarchical linear modelling (multi-level modelling).
By representing the capacity and the demand as uniform random variables, an analytic solution is obtained for the probability distribution of the factor of safety and its probability density and cumulative distribution functions.
By representing the capacity and the demand as independent piecewise linear random variables, an analytic solution is obtained for the probability density and cumulative distribution functions of the factor of safety.
This is mainly because internal forces are changing as the changes of design variables, and these changes don't be reflected in RSM equation, which lead to the difference in the form between RSM approximation with reciprocal variables and analytic solution.
Covariates were divided into the three general groups of related variables for analytic purposes: tumor characteristics and treatment, demographic characteristics, and health characteristics in an approach similar to that of Ganz et al [ 6].
In those analyses, the response variables, data analytic strategy and missing data strategy were coded as binary (0='No', 1='Yes'), with the year of publication as predictor of a 'Yes' response.
To evaluate the relative importance of the subtotals on the global level of satisfaction, a total score was obtained by summing up the scores on all the items, recoding some variables for analytic purposes to ensure that all the items had the same positive direction.
