Basic concepts and theorems of the evolutionary genetics of free populations

It is well known that the principles of biological inheritance, initiated by Mendel in 1865, allow of an exact mathematical formulation. For this reason classical genetics can be regarded as a mathematical discipline.
This article is concerned with the direction in mathematical genetics that stems from the widely known papers of Hardy and Weinberg (1908). It scarcely touches upon purely probabilistic and statistical questions, but uses probabilities (mean values of frequencies) as state coordinates in an “infinitely large” population. Change of state (evolution) occurs under the action of a certain quadratic operator. The paper has two aspects: 1) the structure of free populations; 2) the behaviour of trajectories. The fundamental investigations on these problems were carried out by S. N. Bernstein (1923–1924) and Reiersol (1962). Certain additional results directed towards completing the theory have been found recently by the author and are published here for the first time.
At the beginning of the paper we give a short sketch of the basic notions of classical genetics, in essence simply a minimal glossary. The reader who is familiar with the elements of genetics to the extent, for example, of the popular tract of Auerbach [1] or the appropriate chapters of the textbook by Villee [2], could omit this sketch. For a deeper study of the biological material the books of McKusick [3], Stern [4] and Mayr [5] are recommended.
The elementary mathematical questions of genetics are concerned with certain guiding principles in probability theory (see, for instance, [6]–[8]). The textbooks and monographs [9]–[15] are devoted to mathematical genetics. The sources listed here apply but little to the problems of the present work.
The main results are concentrated in §§ 4, 5, 9, 11. The remaining sections play an auxiliary role.