Long- and Mid-Term Variations of the Soft X-ray Flare Type in Solar Cycles

Using data from the Geostationary Operational Environmental Satellites (GOES) spacecraft in the 1-8 \AA\ wavelength range for Solar Cycles 23, 24, and part of Cycles 21 and 22, we compare mean temporal parameters (rising, decay times, duration) and the proportion of impulsive short-duration events (SDE) and gradual long-duration events (LDE) among C- and $\geq$M1.0-class flares. It is found that the fraction of the SDE $\geq$M1.0-class flares (including spikes) in Cycle 24 exceeds that in Cycle 23 in all three temporal parameters at the maximum phase and in the decay time during the ascending cycle phase. However, Cycles 23 and 24 barely differ in the fraction of the SDE C-class flares. The temporal parameters of SDEs, their fraction, and consequently the relationship between the SDE and LDE flares do not remain constant, but they reveal regular changes within individual cycles and during the transition from one cycle to another. In all phases of all four cycles, these changes have the character of pronounced, large-amplitude "quasi-biennial" oscillations (QBOs). In different cycles and at the separate phases of individual cycles, such QBOs are superimposed on various systematic trends displayed by the analyzed temporal flare parameters. In Cycle 24, the fraction of the SDE $\geq$M1.0-class flares from the N- and S-hemispheres displays the most pronounced synchronous QBOs. The QBO amplitude and general variability of the intense $\geq$M1.0-class flares almost always markedly exceeds those of the moderate C-class flares. The ordered quantitative and qualitative variations of the flare type revealed in the course of the solar cycles are discussed within the framework of the concept that the SDE flares are associated mainly with small sunspots (including those in developed active regions) and that small and large sunspots behave differently during cycles and form two distinct populations.Comment: 15 pages, 6 figures. Accepted for publication in Solar Physic

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

Noncommutative Field Theories and (Super)String Field Theories

In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functional, operator, conformal and the half string formalisms. The main technical tools that are used to study conjectured D-brane decay into closed string vacuum through the tachyon condensation are presented. We describe also methods which are used to study the cubic open string field theory around the tachyon vacuum: construction of the sliver state, ``comma'' and matrix representations of vertices.Comment: 160 pages, LaTeX, 29 EPS figures. Lectures given by I.Ya.Aref'eva at the Swieca Summer School, Brazil, January 2001; Summer School in Modern Mathematical Physics, Sokobanja, Yugoslavia, August 2001; Max Born Symposium, Karpacz, Poland, September, 2001; Workshop "Noncommutative Geometry, Strings and Renormalization", Leipzig, Germany, September 2001. Typos corrected, references adde