[en] This paper describes the soliton surfaces approach to the Witten-Dijkgraaf-E.Verlinde-H. Verlinde (WDVV) equation. We constructed the surface associated with the WDVV equations using Sym-Tafel formula, which gives a connection between the classical geometry of manifolds immersed in R ^m and the theory of solitons. The so-called Sym-Tafel formula simplifies the explicit reconstruction of the surface from the knowledge of its fundamental forms, unifies various integrable nonlinearities and enables one to apply powerful methods of the soliton theory to geometrical problems. The soliton surfaces approach is very useful in construction of the so-called integrable geometries. Indeed, any class of soliton surfaces is integrable. Geometrical objects associated with soliton surfaces (tangent vectors, normal vectors, foliations by curves etc.) usually can be identified with solutions to some nonlinear models (spins, chiral models, strings, vortices etc.) [1], [2]. We consider the geometry of surfaces immersed in Euclidean spaces. Such soliton surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η₁₁ = 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. Also, we study an area of surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η₁₁ = 0. (paper)

[en] Full text: Band coniferous forests are located at the right bank of Irtysh river in two oblasts of Kazakhstan - East Kazakhstan and Pavlodar.This is a unique and only forest of this type. Something similar to this natural treasure with climate-regulating, sanitary, soil-protective, water-preserving functions can be found in Canada only. Total area of the band forest comprises 870500 hectares. The forest is mainly presented by pines (Pinus silvestris). These forests are of relict nature and are of great environmental, social and economic value. The band forests located in northern, north-western and western parts of SNTS were subjected several time to radioactive impacts from atmospheric nuclear tests performed at SNTS. Nuclear clouds from 12 ground and 28 atmospheric explosions passed over these territories. Four nuclear tests performed on 29th of August 1949, 29th of July 1955, 7th of August 1962 and 26th of November 1962 resulted in higher radiation dose rates registered on land there. It seems that this particular tests stipulated radioactive contamination of the forests. The first nuclear test performed on 29th of August 1949 resulted in considerable radioactive contamination of the band forests. Contamination was registerd in Novopokrovskij and Beskaragajskij districts of Semipalatinsk oblast as well as in several districts of Altai Territory. The second test that could bring radioactive contamination to the forests was performed on 7th of August 1962 when instead of planned atmospheric explosion, there was achieved surface explosion with comparatively high radioactive contamination of the lands towards Altai Territory. Within the State program ''Forest preservation and expansion of forest in the Republic of Kazakhstan'' there was performed in 2006 a radiological surveying of the lands in pipe forest of near-Irtysh region. There were studied soil and vegetation as well as woods of the band coniferous forests. Part of territory, wherethrough nuclear clouds went, was investigated along six profiles spacing 2 kilometers between points of investigation. Remaining territory of forests was investigated by the use of small-scale net spacing 10x20 km. Exceeding concentrations of radionuclides in the environment has been revealed on the lands limited by: from the North - Kurchatov - Malaya Vladimirovka, and from the South - Chagan - Dzhelandy. Specific activity of radionuclides in soil lies within the limits: ¹³⁷Cs < 1-3.8*10², ⁹⁰Sr <1-1.2*10², ^239+240Pu < 1-3.6*10² Bq/kg what exceeds the background values for 25, 18 and over 100 times, respectively. The majority of newly introduced radionuclides remain in the top 10 cm of the soil (about 80-90%). At the same time, major accumulation of the radionuclides has been registered in the forest litter. Maximal values of specific activity achieve the following numbers: ¹³⁷Cs - 6.3*10², ⁹⁰Sr - 2*10² Bq/kg. Exceeding contents of ¹³⁷Cs, ⁹⁰Sr, ^239+240Pu at investigated territories were mainly registered within radioactive fallout traces of the first nuclear explosion. No exceeding over background values has deen registered on other lands