Divergences in maximal supersymmetric Yang-Mills theories in diverse dimensions

The main aim of this paper is to study the scattering amplitudes in gauge field theories with maximal supersymmetry in dimensions D=6,8 and 10. We perform a systematic study of the leading ultraviolet divergences using the spinor helicity and on-shell momentum superspace framework. In D=6 the first divergences start at 3 loops and we calculate them up to 5 loops, in D=8,10 the first divergences start at 1 loop and we calculate them up to 4 loops. The leading divergences in a given order are the polynomials of Mandelstam variables. To be on the safe side, we check our analytical calculations by numerical ones applying the alpha-representation and the dedicated routines. Then we derive an analog of the RG equations for the leading pole that allows us to get the recursive relations and construct the generating procedure to obtain the polynomials at any order of (perturbation theory) PT. At last, we make an attempt to sum the PT series and derive the differential equation for the infinite sum. This equation possesses a fixed point which might be stable or unstable depending on the kinematics. Some consequences of these fixed points are discussed.Comment: 43 pages, 13 figures, pdf LaTex, v2 minor changes and references adde

Methodological approach for developing online courses: a case study

Online courses are becoming more popular and accessible, but they require careful design and development of their structure and content. This paper presents a methodological approach for creating online courses by analysing existing practices and learner feedback. The paper focuses on the case study of the online course “Methods for Teaching Mathematics to Students in Technical Universities”, developed and tested by the authors. The paper discusses the main aspects of online course development, such as weekly planning, testing subsystem, video and PDF content, forum and survey subsystems, and assessment of learning outcomes. The paper also evaluates the quality of the online course and identifies some areas for improvement. The paper concludes that online course quality depends on various factors and provides recommendations for online course developers

Nonlinear interfacial waves in a constant-vorticity planar flow over variable depth

Exact Lagrangian in compact form is derived for planar internal waves in a two-fluid system with a relatively small density jump (the Boussinesq limit taking place in real oceanic conditions), in the presence of a background shear current of constant vorticity, and over arbitrary bottom profile. Long-wave asymptotic approximations of higher orders are derived from the exact Hamiltonian functional in a remarkably simple way, for two different parametrizations of the interface shape.Comment: revtex, 4.5 pages, minor corrections, summary added, accepted to JETP Letter

COMBINATIVE ABILITY BY WEIGHT OF 1000 SEEDS OF WINTER WHEAT CULTIVARS WITH WHEAT-RYE TRANSLOCATIONS OF 1AL/1RS

In F1 diallel crosses the effect peculiarities of general and constant of specific combinational ability and their ratios within winter bread wheat, carriers of wheat-rye translocations and without them, according to index «weight of 1000 seeds». The sources of steady high general and specific combinatve abilities are distinguished: Smuhlyanka and Kryzhynka – the carrier of different wheat-rye translocations. The mentioned above cultivars are recommended to be involved in hybridization under conditions of steppes of Ukraine to improve the studied feature. Particular attention should be paid to the cultivar Smuhlyanka – carrier of 1AL/1RS translocation, which represents the largest selection values to create forms with high weight of 1000 seeds whereas it showed the best indicators of general and specific combinative abilities steady for years of study