Hence, if one finds an explicit function satisfying all the properties in Proposition 1, it is the one representing the projected wavefunctions.
In any case, in order to conduct these studies, we first of all have to find out what are the explicit functions representing the projected wavefunctions.
Now we are ready to propose an explicit function
for the effective diffusion coefficient, formally given by Eq 3.
Beyond its explicit function
as an introduction to the papers that follow, it explores some of the most important musical concerns addressed by the new techniques, the author drawing on what will be for many readers very familiar examples from his own compositional work.
It remains to be seen, however, whether explicit equivalence training without explicit function
training will also fail to produce the predicted performances on the final critical test set.
In Section V, we demonstrate how, once an explicit function
for utility is assumed, the expected utility locus can be transformed into an estimable equation which contains the monetary loss equivalent and other utility function parameters.
This indicates that the critical buckling load and coefficient can also be derived as explicit functions
However, all these processing methods can be generally divided into two major research directions, namely: a) design and synthesis of pseudo-NLFM (piecewise) waveforms which are in fact, signals with LFM predistortioned on short intervals into temporal domain or corrected into spectral domain [10,11]; b) design and synthesis of proper (i.e., as predefined shape of the energy/power spectral density (E/PSD) function) pure NLFM waveforms using for example, proper iterative methods [3, 11], stationary phase principle, [12-15] or explicit functions
cluster method  etc.
We will show that there is a real difference in the computational power of k players and k +1 players in this model, by constructing explicit functions
which are easy for k +1 players, and hard for k players.
For some pseudo-multivariate series, we might not be able to have the explicit functions
for the sum of their projection series of one variable.
(13) The model coefficients, further, are explicit functions
of the primitive parameters, including the frequency of price adjustment and the fraction of firms that are backward looking.
Users can plot explicit functions
and parametric equations in both Cartesian and polar coordinates, and equations can be represented as graphs and tables.