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For the pairs of quasi-commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators.

The strong and weak properties of peas and beans cultivars were identified in field experiments. The best productivity of inoculated plants was due to the resistance to biotic and abiotic stress induced by endophytic bacteria.

Let M n ( F ) be the algebra of n × n matrices over the field F and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M n ( F ) cannot exceed 2 n − 2 . We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2 n − 2 is achieved.

Let M n ( F ) be the algebra of n × n matrices and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dim F ⁡ ( C ) ≤ 3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when char F > 2 (see the examples in [1] ), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4.

A sharp upper bound for the length of an algebra as a function of the index of nilpotency of its Jacobson radical and the length of the quotient algebra by the radical is obtained.

We study the length of constructions such as a quotient algebra, a direct sum of algebras, and a subalgebra, which occur in the classical Birkhoff theorem. An upper bound for the length of local algebras is obtained.

The effect of the addition of 18-crown-6 to the water–methanol and water–acetonitrile mobile phase on the selectivity of the separation of different derivatives of phenol, aniline, and benzoic acid by reversed-phase high-performance liquid chromatography was studied. It was demonstrated that the retention of compounds with the protonated primary amino group is increased in the presence of 18-crown-6 in the eluent, which can be explained by complexation between analyte compounds and macrocycles. For compounds containing the hydroxyl group (phenol and its derivatives), the addition of 18-crown-6 to the mobile phase leads to some decrease in retention. It was demonstrated that the selectivity of 18-crown-6 can change depending on the nature of the positional isomer (ortho, meta, orpara) of the analyte and the nature of the solvent increasing or decreasing the complexing ability of the sorbate with the macrocycle. The effect of the concentration of crown ether on the retention of compounds with the protonated amino group was revealed, and the complexation mechanism was suggested.

The modification of active sites of a sorbent containing surface NH2 groups by the addition of 18-crown-6 to the acidified mobile phase (acetonitrile) was studied. A relationship between the chemical structure of modified anion-exchange sites of the stationary phase and the selectivity of ion exchange in the determination of oxygen-containing benzo and dibenzo macrocycles (from 12 to 30 atoms in a cycle) and organic compounds of different classes was revealed.

The use of crown compounds (18-crown-6, benzo-18-crown-6, dibenzo-18-crown-6, benzo-15-crown-5, dibenzo-24-crown-8, 4,13-diaza-18-crown-6,4,13-dibenzyl-diaza-18-crown-6, and cryptand [2.2.2]) as components of stationary phases in the determination of organic compounds by gas chromatography is studied. The polarity, selectivity, efficiency, and working temperature ranges of stationary phases based on crown ethers and cryptand as well as conventional stationary phases for gas chromatography are compared. The influence of the type and number of heteroatoms, the conformational lability of the cavity, and the presence of substituents on the polarity and selectivity of the stationary phases under study is revealed. Preferable types of interactions of stationary phases containing crown compounds with organic compounds of different classes are discussed. It is concluded that phases of the mixed type based on crown ethers are promising for improving the selectivity and efficiency of gas-chromatographic separation

It is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion. The examples are given showing that the converse does not hold. To establish this result, we prove several fundamental properties of the length function.