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

СРАВНИТЕЛЬНАЯ ОЦЕНКА ЭФФЕКТИВНОСТИ ВЫПОЛНЕНИЯ ФОРАМИНОТОМИИ РАЗЛИЧНЫМИ ХИРУРГИЧЕСКИМИ МЕТОДАМИ В ЛЕЧЕНИИ КОРЕШКОВОГО БОЛЕВОГО СИНДРОМА У БОЛЬНЫХ С КОМБИНИРОВАННЫМ ЛАТЕРАЛЬНЫМ СТЕНОЗОМ НА ПОЯСНИЧНОМ УРОВНЕ: ПРОСПЕКТИВНОЕ ИССЛЕДОВАНИЕ

Objective: to compare the therapeutic possibility of the decompressiveviedecompressive with stabilization surgeries using the standard posterior and anterior retroperitoneal approaches in patients with combination of inherent and obtaining lateral stenosis and to demonstrate the adequacy of using. At the main group we removed the herniated disc with stabilization using anterior and posterior approaches – 82 patients. The control group was treated by standard microdiscectomy – 40 patients. More excellent and good results were in the main group where decompression was combined with stabilization, and at the same group were less negative results. The decompressive-stabilizing surgery with anterior interbody fusion is a pathogenetic and technically adequate treatment for combined lateral stenosis.Цель исследования – сравнить клиническую эффективность декомпрессивных и декомпрессивностабилизирующих вмешательств, выполняемых из передних и задних доступов в лечении корешкового болевого синдрома у пациентов с комбинированным латеральным стенозом на поясничном уровне, доказать адекватность применения передних декомпрессивно-стабилизирующих вмешательств. Декомпрессивно-стабилизирующие вмешательства, дифференцированные по способу доступа, выполнены 82 пациентам основной группы, декомпрессивные – 40 больным контрольной группы. В группе с декомпрессивно-стабилизирующими вмешательствами в отдаленном периоде значимо преобладают отличные результаты, в ней же меньше и неблагоприятных исходов. Применение передних декомпрессивно-стабилизирующих вмешательств является адекватным методом хирургической коррекции латерального стеноз

THE COMPARATIVE ASSESSMENT THE EFFECTIVENESS OF TREATMENT THE NERVE ROOT COMPRESS SYNDROME USING THE ANTERIOR AND POSTERIOR APPROACHES OF PATIENTS WITH COMBINED LATERAL LUMBAR STENOSIS

Objective: to compare the therapeutic possibility of the decompressiveviedecompressive with stabilization surgeries using the standard posterior and anterior retroperitoneal approaches in patients with combination of inherent and obtaining lateral stenosis and to demonstrate the adequacy of using. At the main group we removed the herniated disc with stabilization using anterior and posterior approaches – 82 patients. The control group was treated by standard microdiscectomy – 40 patients. More excellent and good results were in the main group where decompression was combined with stabilization, and at the same group were less negative results. The decompressive-stabilizing surgery with anterior interbody fusion is a pathogenetic and technically adequate treatment for combined lateral stenosis

State aid in the Energy Sector: Aid Recovery

A review of the epidemiology of clinical rubella in the Perm region of the Russian Federation from 1979-97 showed that the incidence was about 220 cases per 100,000 population. Congenital rubella syndrome (CRS) accounted for 15% of birth defects and for about 3.5 cases of CRS per 1000 live births per year. Surveys of the seroepidemiology of rubella infection revealed that the susceptibility rate among pregnant women (i.e. rubella virus antibody haemagglutination-inhibition (HAI) assay titres < 10) was 16.5%. As serum rubella antibody HAI titres > or = 10 both prevented infection in pregnant women and protected their foetuses, serological testing has been introduced into the routine antenatal services. Pre-existing rubella antibodies were found not to interfere with the immune response to vaccination, so selective immunization was provided to girls approaching puberty and to women of childbearing age. A programme of epidemiological surveillance is being developed to define tactics for the widescale introduction of rubella vaccination

Ultrafast all-optical control of the magnetization in magnetic dielectrics

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