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In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.

The matrix relation AB = CBA is investigated. An explicit description of the space of matrices B satisfying this relation is obtained for an arbitrary fixed matrix C and a diagonalizable matrix A. The connection between this space and the family of right annihilators of the matrices A−λC, where λ ranges over the set of eigenvalues of the matrix A, is studied. In the case where AB = CBA, AC = CA, and BC = CB, a canonical form for A,B,C, generalizing Thompson’s result for invertible A,B,C, is introduced. Also bounds for the lengths of pairs of matrices {A,B} of the form indicated are provided. Bibliography: 26 titles.

A lower and an upper bounds for the length of a direct sum of nonassociative algebras are obtained, and their sharpness is established. Note that while the lower bound for the length of a direct sum in the associative and nonassociative cases turns out to be the same, the upper bound in the nonassociative case significantly exceeds its associative counterpart.

Relevance. Tooth splints used to immobilize lower jaw fragments lead to the progression of periodontitis.Purpose. To study changes in periodontal tissues in patients with fractures of the lower jaw during immobilization with dentitional double jaw splints.Materials and methods. Patients with periodontitis during the treatment of fractures of the lower jaw were examined.Results. In patients with periodontitis who have a fracture of the lower jaw, inflammatory and destructive processes in periodontal tissues progress during the immobilization of fragments by the orthopedic method (tooth splints).Conclusion. Comprehensive treatment of periodontitis in fractures of the lower jaw, including dynamic magnetotherapy (DMT) and the treatment of periodontal pockets with liniment of cycloferon, reduces the severity of inflammatory and destructive processes in periodontal tissues.

The paper evaluates the lengths of group algebras of all groups of orders not exceeding 7 over an arbitrary field.

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

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 effect of endophytic bacterial associations on the productivity of bean cultivars was studied. It was shown that the efficiency of inoculation with microorganisms depended on the cultivar of beans and climatic environmental factors.

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.

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.

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.

We describe matrices with extremal generalized centralizers over algebraically closed fields.

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N n (𝔽) over a field 𝔽 with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M n (𝔽). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.

The paper continues the investigation of the lengths of quasi-commuting matrix pairs; specifically, it considers the problem of realizability of different positive integers as values of the length function for quasi-commuting matrix pairs.

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.

The notion of graph generated by the mutual orthogonality relation for the elements of an associative ring is introduced. The main attention is paid to the commutative rings and to the matrix ring over a field and its various subrings and subsets. In particular, the diameters of the orthogonality graphs of the full matrix algebra over an arbitrary field and its subsets consisting of diagonal, diagonalizable, triangularizable, and nilpotent matrices are computed. Bibliography: 36 titles.

A matrix is said to be semimagic if all its row and column sums are equal. The paper investigates the length function for the algebra of semimagic matrices with respect to different generating systems. Bibliography: 11 titles.

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.

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean the least positive integer k such that words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, we study the main ring-theoretical properties of the length function: the behavior of the length under unity adjunction, direct sum of algebras, passage to subalgebras and homomorphic images. We give an upper bound for the length of the algebra as a function of the nilpotency index of its Jacobson radical and the length of the quotient algebra. We also provide examples of length computation for certain algebras, in particular, for the following classical matrix subalgebras: the algebra of upper triangular matrices, the algebra of diagonal matrices, the Schur algebra, Courter’s algebra, and for the classes of local and commutative algebras.

We define the length of a finite system of generators of a given algebra \( \mathcal{A} \) as the smallest number k such that words of length not greater than k generate \( \mathcal{A} \) as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length 1 up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length 1, as well as those that are maximal with respect to inclusion.

Linear algebraic version of celebrated Double Centralizing Theorem states that the set of matrices commuting with all matrices from a centralizer of a given matrix [Formula: see text] coincides with the set of polynomials in [Formula: see text]. We examine the existence of an analogue of this classical result once commutativity is substituted by commutativity up to a factor, which is an important relation in quantum physics.

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.

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer k such that the words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.

Let {ie910-01} be a field and let {ie910-02} be a finite-dimensional {ie910-03}-algebra. We define the length of a finite generating set of this algebra as the smallest number k such that words of length not greater than k generate {ie910-04} as a vector space, and the length of the algebra is the maximum of the lengths of its generating sets. In this article, we give a series of examples of length computation for matrix subalgebras. In particular, we evaluate the lengths of certain upper triangular matrix subalgebras and their direct sums, and the lengths of classical commutative matrix subalgebras. The connection between the length of an algebra and the lengths of its subalgebras is also studied.

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

The data on the immune and autoimmune disturbances in certain brain diseases such as brain tumors, remote consequences of craniocerebral injury, epilepsy, and postradiation encephalopathy are presented. Various changes in cellular and humoral immunity are established. The development of autoimmune reactions to brain antigens is analyzed in comparison with the level of immune complexes. The results are discussed in the context of disruption of the immune system neurohumoral regulation and development of neurogenic immunity deficiency and its different types. The specific function of the immunocompetent cells, which penetrated beyond the blood-brain barrier under physiological and immunopathological conditions of the CNS, and the apoptotic death of the lymphocytes were suggested. Attention was drawn to the importance of the investigation of systemic, as well as regional, immune processes outside the blood-brain barrier.

Mutations in the rpoB, katG, inhA, oxyR/ahpC genes in rifampicin- and isoniazid-resistant M. tuberculosis strains isolated from residents of Moscow, Astrakhan', and Moldova Republic were studied by molecular biological methods (heteroduplex analysis, single strand conformational polymorphism, biochips). Twenty-five combinations of mutations were detected. Some differences in the type distribution of detected mutations were found. The use of biochips is the most perspective method for determining the type of mutation.

The resistance of Mycobacterium tuberculosis (MBT) to fluoroquinolones is associated with the mutations concentrated in the gyrA gene that is a structural gene of a gyrase A subunit. Detection of mutations in this portion of the gene allows the sensitivity of MBT to this group of drugs to be rapidly determined.

We studied the immunocorrective effect of the allogenic new-born brain cells on the model of rats experimental allergic encephalomyelitis (EAE). EAE was induced by the immunization of old rats spinal cord homogenate in Freund's adjuvant. Correction was carried on the 12-th, 14-th, 16-th day after the induction of the EAE by the intraperitoneum injection of rat's new-born brain fractions enriched with neurons and glial cells. A positive clinical effect was achieved by the employment of neurons (the stabilization of encephalomyelitis and the acceleration of the recovery). The glia correction was accompanied by the aggravation in the course of encephalomielitis and by extension of its clinical manifestation period. The obtained results testify to the existence of both an immunoregulative and a neurotrophic influence of the neuron fraction of the new-born brain cells. The mechanism of a corrective effect needs further special investigation.

The influence of the positively charged amphiphilic compound cetyltrimethyl ammonium bromide (CTAB) on palmitate- and laurate-induced uncoupling and on carboxyatractylate and glutamate recoupling effects in liver mitochondria have been studied. CTAB (40 microM) in the presence of 3 mM MgCl2 had little (if any) effect on the palmitic acid-stimulated respiration of mitochondria; the glutamate recoupling effect increased, and the carboxyatractylate recoupling effect decreased to the same degree with the combined effect (about 80%) remaining unchanged. Thus, CTAB decreases the ADP/ATP antiporter involvement and increases to the same extent the aspartate/glutamate antiporter involvement in the fatty acid-induced uncoupling. The carboxyatractylate and glutamate recoupling effects were less pH dependent in the presence of CTAB than in its absence. These data could be interpreted with the assumption that fatty acid anions are more accessible to the ADP/ATP antiporter and their neutral forms are more accessible to the aspartate/glutamate antiporter, and that CTAB changes the relative anion carrier involvement in the fatty acid-induced uncoupling as it forms neutral complexes with fatty acid anions.

Commutative subalgebras of length n − 1 in the full matrix algebra of order n over an arbitrary field are characterized in terms of generating elements.

Measurements of parathyroid hormone concentrations were made in the serum of patients with vestibular larynx before treatment, after radiation treatment and after radiation treatment followed by surgery. Patients with laryngeal cancer were found to have impaired calcium metabolism, their serum parathyroid hormone concentrations were elevated. After radiotherapy, in a favorable clinical course, patients with cancer of the vestibular larynx stage II show normalization of the hormone levels, whereas the above combined treatment diminished the concentrations, but they still remain above normal.

Study of the uncoupling effect of various saturated fatty acids (from caprylic to palmitic) revealed that the glutamate recoupling effect was more pronounced in the case of short chain fatty acids, whereas recoupling of mitochondria by carboxyatractylate was more effective in the case of long chain fatty acids. The overall recoupling effect, however, did not depend on the fatty acid chain length. Besides carboxyatractylate, glutamate and aspartate also exhibited a recoupling effect under uncoupling by lauryl sulfate. The uncoupling effect of lauryl sulfate was markedly weaker in the presence of DNP or laurate (but not FCCP) which were added in concentrations causing twofold increase in mitochondrial respiration. In the presence of lauryl sulfate the uncoupling action of laurate and DNP was insensitive to carboxyatractylate and glutamate. With laurate and DNP as uncouplers increasing the pH from 7.0 to 7.8 potentiated the recoupling effect of carboxyatractylate and attenuated the recoupling effect of glutamate. In the case of uncoupling by lauryl sulfate similar changes in the recoupling effect of carboxyatractylate and glutamate were observed only in the presence of 10 microM tetraphenylphosphonium. Thus, when uncoupling is induced by fatty acids, DNP, and lauryl sulfate, the ADP/ATP and aspartate/glutamate antiporters function as two parallel and independent pathways for mitochondrial membrane potential dissipation. We suggest that the role of the ADP/ATP antiporter in uncoupling includes proton capture from the intermembrane space with subsequent protonation of uncoupler anions, their transport as neutral molecules on the internal side, and deprotonation followed by proton release into the matrix and transfer of the uncoupler anion in the reverse direction. During uncoupling the aspartate/glutamate antiporter cyclically carries the uncoupler anion with simultaneous proton transfer from the intermembrane space into the matrix.

Effects of dicarboxylic fatty acids with varying positions of the carboxyl groups on respiration and membrane potential of liver mitochondria were studied. Tetradecylmalonic acid (a fatty acid with two carboxyl groups in the alpha-position) efficiently uncoupled oxidative phosphorylation similarly to palmitic acid with the same number of carbon atoms. Similarly to the uncoupling by palmitic acid, the coupling effects of carboxyatractylate and glutamate changed reciprocally with changes in pH of the incubation medium: on increasing the pH from 7.0 to 7.8, the coupling effect of carboxyatractylate increased and that of glutamate decreased. A dicarboxylic fatty acid with the second carboxyl at the end of the alkyl chain in the omega-position (alpha, omega-tetradecyldicarboxylic acid) stimulated respiration of the mitochondria at a significantly higher concentration than myristic acid with the same number of carbon atoms, but unlike the latter nearly failed to decrease the transmembrane potential DeltaPsi. Neither carboxyatractylate nor glutamate inhibited the respiration stimulated by this dicarboxylic fatty acid.

Radioimmunoassay was employed to investigate time course changes in blood serum aldosteron in 28 patients with cancer of the middle laryngeal part before and after radiation treatment as well as after radiation treatment followed by surgical intervention. Initial values of aldosteron were low evidencing impaired mineral-corticoid activity of the adrenal cortex. Aldosteron returned to normal after effective radiotherapy at early tumor stage and after radical surgical treatment at later cancer stages. Hormonal unbalance provides more significant information on tumor involvement of the larynx and is a good indicator of the treatment efficiency.