# Selected results in mathematics

*A matrix A of order n is said to be T-factorable if there exists a lower-triangular matrix B and an upper-triangular matrix C such that A = BC.*

Kirzhner, V.M.(*Kirzner, V.M.*) (1965).
**A criterion for the representability of a matrix in the form
of a product of a lower-triangular and an upper-trianglar
matrix.** *Uspehi Math. Nauk* (*Russian Mathematical Survey*) **XX** 1965,6 102-104.(in Russian). (Abstract) (Article.tfactor in russian) .

- The natural number q is called the critical exponent of the norm in the n-dimentional spase (real or complex), if

1. For all linear operators A in n-dimentional space with ||A||=1 ||A^q||=1 => ||A^m||=1 (m>q). 2. There exists a linear operator Ao (||Ao||=1) such that ||Ao^q-1||=1, ||Ao^q||<1.

- Main result

**Theorem 1.** If the unit sfere S={x:||x||=1} can be embedded in an algebraic variety F not containing the origin, ythen the critical exponent exist.

Kirzhner V., Tabachnicov M.I .(1971). **On the critical exponents of norms in an N-dimensional space.** *Siberian Mathematical Journal*., **XII**, 3, 480-483 (in Russian). (English translation: *Sibirskii Mathematicheskii Zhurnal*, 12, 3, 672-675)(Article.critdeg)

*The solution of the problem of optimal subdividing of finite automat on states for the effective detection of malfunctions.*

Kirzhner V., Tabachnicov M.I. (1971). **On a problem of optimal choice.**
Proceeding Low Temperature Physic Institute, *"Computing Mathematics & Computers"*, **II**, 20-25, (in Russian).

*An analysis of some heuristic algorithms.*

Kirzhner V., Rublinecky V.I. (1973). **On the "go to the nearest city" strategy in the traveling salesman problem.** Proceeding Low Temperature Physic Institute *"Computing Mathematics& Computers"*, **IV**, 40-41 (in Russian).

*The upper estimate of the shortest tour of a traveling salesman in terms of the average tour and the dispersion of tour lengths.*

Kirzhner V., Rublinecky V.I. (1974). **An upper limit for the traveling salesman minimal tour.** Proceeding Low Temperature Physic Institute *"Computing Mathematics & Computers"*, **V**, 120-122 (in Russian).

*The well-known Prim-Kruskal and Steiner problems are generalized for a dynamic case when nodes of the communication network appear at pre-determined moments and all the network has a finite lifetime. It is shown that a duynamic version of even prim-Kruskal problem is an NP-complete one. Some useful theorems are proved and three algorithms are suggested to solve the dynamical problem.*

Zaycev I.., Vayner V.., Kirzhner V. (1988). **A dynamic problem of connecting objects with a network.** *Ukrainian Doklady*, A, 6, 70-73 (in Russian).

*Estimates of the solution for the optimization NP-complete problem are obtained by computing of the mean value of the functional, together with its variance. This approach is applied to the Steiner problem in the discrete and continuous forms as well as to a dynamic problem for constructing the developing communication networks.*

Zaycev I.., Vayner V.., Kirzhner V. (1988). **An estimate of the functional optimum in problems of connecting objects with a network.** *Ukrainian Doklady*, Ser. A, 8, 71-74.(in Russian).