### Article

## Preface "Sea hazards"

Sea hazards

Results of the analysis of long time series of sea level for June - September, 2009 near the Aniva cape (Sakhalin), recorded by pressure bottom station placed on depth of 12 m are given. There are 394 abnormal big waves, waves satisfying to freak wave amplitude criterion (the wave height exceeds the significant height more then twice) recorded. The amplification during six events exceeds 2.5 times. The cumulative frequency of the abnormal wave appearance is described by the Poisson distribution as follows from the theory of extreme statistics. Freak waves occurred on the average twice a day which is in good comparison with the Raleigh prediction for narrow-band Gaussian wave field.

We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrodinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed by the NLS.

In this work, we consider the problems of job flow distribution and ranked job framework forming within a model of cycle scheduling in Grid virtual organizations. The problem of job flow distribution is solved in terms of jobs and computing resource domains compatibility. A coefficient estimating such compatibility is introduced and studied experimentally. Two distribution strategies are suggested. Job framework forming is justified with such quality of service indicators as an average job execution time, a number of required scheduling cycles, and a number of job execution declines. Two methods for job selection and scheduling are proposed and compared: the first one is based on the knapsack problem solution, while the second one utilizes the mentioned compatibility coefficient. Along with these methods we present experimental results demonstrating the efficiency of proposed approaches and compare them with random job selection.

The properties of extreme wave storms in the Darss Sill area, SW Baltic Sea, are analysed based on waverider data for 1991-2010 and long-term numerical simulations. The long-term significant wave height is HS ~0.7 m and the most frequent wave periods 2-4 s. The largest measured HS is 4.46 m. The typical measured and modelled wave periods differ by up to 2 s. The annual maximum HS has notched behaviour, with an increase for 1958-1990 and since 1993, and a drastic decrease in 1991-1992. The measured annual average and maximum HS have changed insignificantly in 1991-2010 but the threshold for the top 1% of waves has considerably decreased.

In this work, we propose approaches to creation of a ranked jobs framework within a model of cycle scheduling in virtual organizations of utility Grids with the decoupling of users from resource providers. Two methods for job selection and scheduling are proposed and compared: the first one is based on the knapsack problem solution, while the second one introduces a heuristic parameter of a job and a computational resource set “compatibility”. Along with these methods we present experimental results demonstrating the efficiency of proposed approaches and compare them with random job selection.

The book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achevements in this area obtained in recent times. The reader of this book need to be familiar only with basic courses in differential equations and smooth manifolds.

In this work, we describe approaches to creation of a ranked jobs framework within the model of cycle scheduling in Grid virtual organizations with such quality of service (QoS) indicators as an average job execution time and a number of required scheduling cycles. Two methods for job selection and scheduling are proposed and compared: the first one is based on the knapsack problem solution, while the second one introduces a heuristic parameter of job and computational resources “compatibility”. Along with these methods we present experimental results demonstrating the efficiency of proposed approaches and compare them with random job selection.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.