### Article

## Equilibrium Kawasaki dynamics and determinantal point processes

Investigations of dynamic properties of complex nonlinear systems are a very promising area of modern science. Information about behavior of such system is often fixed as multidimensional point process where every event is a point in a space-time volume. If a time of observation is limited and a number of events in the dataset is no more than several tens the process can be perceived by an external observer as a random even if this process is due to the actions of a few regular factors or the interactions between of the elements of the complex system. A very small number of events, adding a certain amount of truly random events, fluctuations in time and location of the events are leads to the ineffectiveness of traditional methods of processing noisy signals.

This paper presents a method for the analysis of multivariate point processes based on a given physical model of the interaction of the system elements and \ or the joint effect of several external factors that have a regular character. One-dimensional model of a point process was analyzed in detail, the possibility of generalization of the proposed method to the multidimensional case was showed. The main idea of the method is to identify in the event flow quasi-periodic chains of events characterized by a high degree of non-randomness. A periodic slit mask (similar to Dirac comb) was used to find the subset of events. Of all found chains were chosen only chains with the non-randomness level of above the some threshold.

Artificially generated point process was used to compare of the proposed method with the method based on an assessment of the harmonic modulation degree of the Poisson point process intensity. Test event occurs if at least three of the six harmonic oscillations with different phases and periods overlap near to zero level. This dataset was consisted of 79% and 21% deterministic and random events respectively. It was found that the traditional method failed to reveal five of the six periodic components that have been used for generating test point process. In contrast, the proposed method based on estimating the degree of non-randomness of quasi periodic chains reliably detects all the original periodic components. More over, it was shown that the proposed method provides a much better sensitivity and resolution. Errors in determining of test harmonic oscillation periods were less than 0.057%. Note that along with real components in the spectrum of the point process is fixed many other peaks are associated with the fractional multiples of the periods of quasi-periodic chains. These peaks are similar to harmonics in the standard spectral analysis. Availability of information about events that are part of quasi-periodic chains and high resolution presented method can accurately detect and identify such fractional multiple harmonics.

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional. Theorem 4.1, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.

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.