Anti-commutative Groebner-Shirshov basis of a free Lie algebra

One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962)

Composition-Diamond lemma for λ\lambda-differential associative algebras with multiple operators

In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K$ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential Rota-Baxter algebras. In particular, linear bases of free $\lambda$-differential Rota-Baxter algebras are obtained and consequently, the free $\lambda$-differential Rota-Baxter algebras are constructed by words

Bioinformatic technology in the preparation of road transport professionals

Тhe article talks about information technologies in training specialists of automobile transport. Given the definition of and boundaries. Specified the positive aspects of these technologies in education. The role of case studies in trainingВ статье говорится о биоинформационных технологиях в обучении специалистов автомобильного транспорта. Даются определения понятия и границы его использования. Уточняются положительные стороны этих технологий в образовании. Раскрывается роль кейс-стади в подготовке специалисто

Groebner-Shirshov basis for HNN extensions of groups and for the alternating group

In this paper, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Groebner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained

Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

Radiofrequency pulse modulation of the penile dorsal nerve for the treatment of premature ejaculation: evaluation of effectiveness

Introduction.  Premature ejaculation (PE) is the second most common male sexual dysfunction with a significant negative impact on quality of life. A significant number of conservative and surgical methods for the treatment of PE are proposed, but their effectiveness and, in some cases, safety, remain the subject of discussion.Objective. To study the effectiveness of pulsed radiofrequency impulse neuromodulation (RPN) with different modes on the penile dorsal nerve in treatment of patients with premature ejaculation.Materials &amp; methods. Two groups of PE-patients participated in a prospective multicenter study: Group 1 — 11 patients aged 22 – 34 years; Group 2 — 12 patients aged 23 – 34 years. Group 1 received pulsed RPN at a temperature of 42°C, a stimulation frequency of 2 Hz, 45 V, and a duration of 180 seconds; in Group 2 — pulsed RPN at a temperature of 50 °C and the same frequency and duration of stimulation as Group 1. All patients before and after treatment measured intravaginal ejaculation latency time (IELT) and completed a premature ejaculation profile (PEP). The results were evaluated after 30 days.Results. In both groups, there was a statistically significant increase in IEDT and PEP. Thus, in Group 1, IELT increased from 58 to 101.9 seconds, PEP from 3.8 to 12.6 points. In Group 2, IELT increased from 66.5 to 210.9 seconds, PEP from 3.2 to 12.6 points. At the same time, in Group 2, IELT increased by 107% more than in Group 1. No significant side effects were observed in both groups.Conclusion. RPN appears to be a very promising treatment for premature ejaculation, as it is a minimally invasive, safe, and reversible procedure. The technique of influencing the penile dorsal nerve at an electrode temperature of 50 °C compared to 42 °C seems to be more effective, however, additional studies are needed

Constructions of free commutative integro-differential algebras

In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gr\"obner-Shirshov bases.Comment: arXiv admin note: substantial text overlap with arXiv:1302.004

БРОНХОЛЕГОЧНОЕ ПОРАЖЕНИЕ У РЕБЕНКА РАННЕГО ВОЗРАСТА, БОЛЬНОГО ТУБЕРКУЛЕЗОМ

Tuberculosis of intrathoracic lymph nodes in part of patients is complicated by the break of molten caseous node in the lumen of the bronchus lumen and obstruct it with thick caseous masses and granulation tissue. The severity of state in children is determined by the caliber of the affected ones. У части детей раннего возраста туберкулез внутригрудных лимфатических узлов осложняется прорывом лимфатического узла с расплавленным казеозом в просвет бронха с развитием обтурации плотными казеозными массами и грануляционной тканью. Тяжесть состояния обуславливается калибром пораженного бронха.