The Russian 5.45×39mm cartridge is a rimless bottlenecked rifle cartridge. It was introduced into service in 1974 for use with the new AK-74 assault rifle. It gradually supplemented then largely replaced the 7.62×39mm round in service.

The 5.45×39mm is an example of an international tendency towards relatively small sized light weight military service cartridges. Cartridges like the Russian 5.45×39mm, 5.56×45mm NATO and Chinese 5.8×42mm allow a soldier to carry more ammunition for the same weight compared to their larger and heavier predecessor cartridges and produce relatively low bolt thrust and free recoil, favouring light weight arms design and automatic fire accuracy.

The Russian original military issue 5N7 cartridge variant introduced in 1974 are loaded with full metal jacket bullets that have a somewhat complex construction. The 3.2 g (49.4 gr) boattail projectile has a gilding-metal-clad jacket. The unhardened steel core is covered by a thin lead coating which does not fill the entire point end, leaving a hollow cavity inside the nose. The bullet is cut to length during the manufacturing process to give the correct weight. The 5N7 uses a boattail design to reduce drag and there is a small lead plug crimped in place in the base of the bullet. The lead plug, in combination with the air space at the point of the bullet, has the effect of moving the bullet's center of gravity to the rear; the hollow air space also makes the bullet's point prone to deformation when the bullet strikes anything solid, inducing yaw. The brown-lacquered steel case is Berdan-primed. Its 39.37 mm (1.55 in) length makes it slightly longer than the 7.62×39mm case which measures exactly 38.60 mm (1.52 in). The primer has a copper cup and is sealed with a heavy red lacquer. The propellant charge is a ball powder with similar burning characteristics to the WC 844 powder used in 5.56×45mm NATO ammunition. The 5N7 cartridge weight is 10.75 g (165.9 gr).

Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.

By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem) that quintic (and higher-order) equations cannot be solved by radicals (1799), and Ruffini's rule which is a quick method for polynomial division. Ruffini also made contributions to group theory in addition to probability and quadrature of the circle.

He practiced as both a professor of mathematics (University of Modena) and a doctor including scientific work on typhus.

Ruffini’s 1799 work marked a major development for group theory. Ruffini developed Joseph Louis Lagrange's work on permutation theory, following 29 years after Lagrange’s "Réflexions sur la théorie algébrique des equations" (1770–1771) which was largely ignored until Ruffini who established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartics. This angered many members of the community such as Malfatti (1731–1807). Work in this area was later carried on by those such as Gauss and Galois who succeeded in such a proof.