The final section addresses the math in appendices on theorems of implicit functions
, the local Lipschitz Constant of entropy operators, zero-order multiplicative algorithms for positive solutions to nonlinear equations, and multiplicative algorithms for positive solutions to entropy-quadratic programming problems.
More difficult topics include sequences and series of functions, Fourier series, functions of several variables, an in-depth exploration of derivatives, implicit functions
and optimization, parametric integration, and integration in Rn.
Since the kernels support multi-level representation, given n-dimensional data can be converted to implicit functions as unified forms with the level of detail.
Once coefficients of the implicit functions for the highest detail level are found, coefficients for the other levels can be simply calculated by Equation (5).
In number of works published before, for definition of a set of singular configurations the authors used methods based on known theorems about implicit functions reduced to the analysis of the manipulator Jacobian (Haug et al.
according to the known theorem of implicit functions (Litvin et al.
In recent years, Popa  utilized implicit functions instead of contraction conditions to prove common fixed point theorems.
We also show that numerous contrastive conditions of the existing literature enjoy the format of our newly introduced implicit function besides admitting several new and natural contrastive conditions.
The set-up of the sub-problem of Pareto-optimal ESS IPS complex correction for instantaneous states in normal conditions, in the scope of main clauses of GMRG and considering the first main clause, in particular at application of the theory of implicit functions
by continuous idealisation of changes of variables, leads to the following multi-objective search:
It can create several types of graphs: Cartesian graphs, graphs of tables, polar, parametric and implicit functions
, inequalities and slope fields.
Generalization of the classical theorems on the existence of inverse or implicit functions
of locally Lipschitz continuous functions are given in Clarke (1983) and Kummer (1991).
i](z) will change now into the implicit functions
of the vector x: [F.