This paper is a brief version of the monography [1]. The m-M Calculus deals
with the so-called m-M functions, i.e. functions of the form $f: D \rightarrow R$
( $D = [a_1, b_1] \times \ldots \times [a_n, b_n]$, where $n > 0$ is any integer
and $a_i, b_i \in R$ ) subjected to the following supposition:For each n-dimensional segment $\Delta = [\alpha_1 , \beta_1] \times \ldots \times [\alpha_n , \beta_n] \subset D$ a pair of real
numbers, denoted by $m(f)(\Delta)$, $M(f)(\Delta)$, satisfying the conditions\begin {equation}
m(f)(\Delta) \leq M(f)(\Delta), \\ \\ (for all \Delta \subset D, X \in \Delta)
\end{equation}
\begin{equation}
\lim_{diam \Delta \to 0}(M(f)(\Delta)-m(f)(\Delta))=0, \\ \\ (where diam \Delta : (\sum(\beta_i - \alpha_i)^2)^{1/2})
\end {equation}
is effectively given.Such an ordered pair $$ of mappings m(f), M(f) (both mapping the set of all $\Delta \subset D$ into R)
is called an m-M pair of the function f. We also say that m(f), M(f) are generalized minimum and maximum
for f respectively. For instance, with only few exceptions all elementary functions are m-M functions (Lemma 1.2).The conditions (1) and (2) are taken as axioms of the m-M calculus. A
logical analysis of these axioms is given here and, in addition to the other results , a
series of equivalences is proved which enable us to express some relationships for m-M
functions by means of the corresponding relationships for their m-M 'pair s (see (2. 2),
(2 .5), (2 .6 ), (2 .7) ). There are many various applications of the m-M calculus, such as
\begin{itemize}
\item Solving systems of inequalities, systems of equations (Section 1)
\item Finding n-dimensional integrals (Section 1, Example 1.5)
\item Solving any problem expressed by a positive $leq$ formula (Section 2), among others
\begin{enumerate}
\item Problem of constrained optimization (Problem 2.2, Problem 2.3)
\item Problem of unconstrained optimization (Problem 2.1)
\item Problems from Interval Mathematics (Problem 2.4)
\end{enumerate}
\item Finding functions satisfying a given m-M condition (e.g, funclional condition, or
difference equation, or differential equation, Section 3).
\end{itemize}As it is well known, by the usual methods of numerical analysis , assuming
certain convergence conditions , we approximately determine, step-by-step, one
solution of the given problem. However, applying the methods of m-M calculus we
approximately determine all solutions of the given problem, and we assume almost
nothing about the convergence conditions . The solutions are, as a rule, sought in a
prescribed n -dimensional segment D. If the given problem, e.g. a system of equations,
has no solutions in D, then applying the method of m-M calculus this can be
established at a certain finite step k. The basic methodological idea of the m-M
calculus is :it gives a sufficient condition $Cond(\Delta)$ which ensures that an n-dimensional
segment $\Delta$ does not contain any solution of the considered problem P. Applying
repeatedly this criterion, we reject from the original n -segment D those "pieces"
which do not contain solutions, so that in the limit case the remaining "pieces"
form the set of all solutions of the problem P (if indeed there is a solution of P).
