### Article

## Note on local mixing techniques for stochastic differential equations

Several methods of establishing coupling for stochastic differential equations are presented.

The idea of code quality assessment is well known for a long time; class connectivity metrics were proposed by community several years ago and have not become generally applicable practice in industrial programming. The objective of the study, part of which we present in this paper, is to critically analyze the metrics available for today: Are they completely unusable, or considering their specifics can be useful helpers for software specialists. Specifically, we try to answer the questions: Are there any connection between design patterns and cohesion metrics, and how these patterns affect metrics if they do.

The content of this volume is mainly based on selected talks that were given at the “International Meeting on Game Theory (ISDG12-GTM2019),” as joint meeting of “12th International ISDG Workshop” and “13th International Conference on Game Theory and Management,” held in St. Petersburg, Russia on July 03–05, 2019. The meeting was organized by St. Petersburg State University and International Society of Dynamic Games (ISDG). Every year starting from 2007, an international conference “Game Theory and Management” (GTM) has taken place at the Saint Petersburg State University. Among the plenary speakers of this conference series were the Nobel Prize winners Robert Aumann, John Nash, Reinhard Selten, Roger Myerson, Finn Kidland, Eric Maskin, and many other famous game theorists. The underlying theme of the conferences is the promotion of advanced methods for modeling the behavior that each agent (also called player) has to adopt in order to maximize his or her reward once the reward does not only depend on the individual choices of a player (or a group of players), but also on the decisions of all agents that are involved in the conflict (game).

This article discusses stochastic numerical methods of Runge-Kutta type with weak and strong convergences for systems of stochastic differential equations in Itô form. At the beginning we give a brief overview of the stochastic numerical methods and information from the theory of stochastic differential equations. Then we motivate the approach to the implementation of these methods using source code generation. We discuss the implementation details and the used programming languages and libraries

We study discrete time totally asymmetric exclusion process (TASEP) describing collective random walk of countable particle configurations in heterogeneous continuum. A typical example of this sort is a traffic flow model with obstacles: traffic lights, speed bumps, spatially varying local velocities etc. Ergodic properties of such systems are studied, in particular we obtain the so called Fundamental Diagram: dependence of average particle velocities on particles and obstacles densities and jump probabilities. The main technical tool is a "dynamical" coupling construction applied in a nonstandard fashion: instead of proving the existence of the successful coupling (which even might not hold) we use its presence/absence as an important diagnostic tool. This techniques allows to reduce the calculation of the average velocity to the similar problem for an auxiliary lattice zero-range process.

Background

The spatiotemporal coupling of brainwaves is commonly quantified using the amplitude or phase of signals measured by electro- or magnetoencephalography (EEG/MEG). To enhance the temporal resolution for coupling delays down to millisecond level, a new power correlation (PC) method is proposed and tested.

New method

The cross-correlations of any two brainwave powers at two locations are calculated sequentially through a measurement using the convolution theorem. For noise suppression, the cross-correlation series is moving-average filtered, preserving the millisecond resolution in the cross-correlations, but with reduced noise. The coupling delays are determined from the delays of the cross-correlation peaks.

Results

Simulations showed that the new method detects reliably power cross-correlations with millisecond accuracy. Moreover, in MEG measurements on three healthy volunteers, the method showed average alpha–alpha coupling delays of around 0–20 ms between the occipital areas of two hemispheres. Lower-frequency brainwaves *vs.* alpha waves tended to have a larger lag; higher-frequency waves *vs.* alpha waves showed delays with large deviations.

Comparison with existing methods

The use of signal power instead of its square root (amplitude) in the cross-correlations improves noise cancellation. Compared to signal phase, the signal power analysis time delays do not have periodic ambiguity. In addition, the novel method allows fast calculation of cross-correlations.

Conclusions

The PC method conveys novel information about brainwave dynamics. The method may be extended from sensor-space to source-space analysis, and can be applied also for electroencephalography (EEG) and local field potentials (LFP).

We strengthen the convergence result in our paper, ibid. 5, No. 6, 1059-1098 (1999; Zbl 0983.62049), proving the local asymptotic mixed normality property in one of the 11 cases considered in that paper.

In this paper we introduce stochastic parameters into the network game model with production and knowledge externalities. This model was proposed by V. Matveenko and A. Korolev as a generalization of the two-period Romer model. Agents differ in their productivities which have deterministic and stochastic (Wiener) components. We study the dynamics of a single agent and the dynamics of a dyad where two agents are aggregated. We derive explicit expressions for the dynamics of a single agent and dyad dynamics in the form of Brownian random processes, and qualitatively analyze the solutions of stochastic equations and systems of stochastic equations.

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.