This spirit had led Professor Saitoh to several fundamental results on the theory of linear transforms, Pythagorean theorems, several very general norm inequalities, representations of non-linear simultaneous equations and implicit functions
, different types of applications of the Tikhonov regularization (including a typical main result on a numerical and real inversion formula of the Laplace transform, with the coauthors Professors Hiroshi Fujiwara and Tutomu Matsuua).
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.
The following theory is a systematic development of all functions covering properties of constrained and unconstrained functions that are continuous and differentiable to various specified degrees [1, 2] and the proof of the existence of implicit functions
 for the form of these functions to be optimized.
In this paper, we assume that the boundary functions of arbitrary obstacles can be known from the implicit functions
which can be constructed from sensor readings or image data.
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., 1996, Abdel-Malek & Yeh, 1997).
In recent years, Popa  utilized implicit functions
instead of contraction conditions to prove common fixed point theorems.
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).
Namely in the complex optimal control theory of big power systems (PS) and interconnected power systems (IPS), this approach has been implemented most widely and efficiently for solving hierarchic sub-problems of optimization within the framework of the so-called Generalized Reduced Gradient Method (GRGM) where, in addition to the implicit functions
theory, a rational choice of a basis is used (i.e., the composition of the components of vectors of dependent and independent parameters) for speeding up the finding of an optimal solution [9-11].