Some new principles (in real analysis, geometry and complex analysis) with consequent two trends in equations

The results of a general nature related to basic concepts in mathematics (usually referred as principles) were established mostly before and during the 19th century and quite few in 20th century. We mean here primarily such a basic concepts as arbitrary enough smooth real functions of one and two variables, plane curves and surfaces in R3, arbitrary meromorphic (particularly analytic) functions in a given domain.

In this talk we present some new results related to the mentioned concepts. Among them:

• principle of zeros of real functions permitting to give bounds for the number of zeros of the functions;
• principle of angles for the plane curves permitting to compare rotations of the curves around a given center and tangential rotations;
• principle of logarithmic derivatives of meromorphic functions permitting to compare different integrals including the logarithmic derivatives.

All these principles are brand new, i.e., have no predecessors.

It is interesting that for all three principles there are some versions similar to the classical Nevanlinna deficiency relation (in complex analysis).

Then we will show how the principles can be applied to some basic equations; among them arbitrary first order differential equations, systems of linear differential equations, Schrödinger equations etc.. This leads to two brand new trends respectively in real and complex differential equations. The principles related to real functions and curves permit to study the zeros and "rotations" of solutions or real differential equations which were not touched before; the principle related to meromorphic functions permits to study a "virgin land" in complex differential equations, namely the zeros of meromorphic solutions of some basic equations in a given domain; previously the zeros were studied only for meromorphic solutions in the complex plane or in the disks.