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## Overlap of two Brownian trajectories: exact results for scaling functions

We consider two random walkers starting at the same time t = 0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d < 4, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable R^2/t. We provide general integral formulas for scaling functions for arbitrary dimensionality d < 4. In contrast, we show that no scaling function exists for higher dimensionalities d more or equal to 4.

This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.

This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probabilities of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that in tegrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.

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 paper contains detailed exposition of the results presented in the short communication [M. V. Zhitlukhin and A. A. Muravlev, Russian Math. Surveys, 66 (2011), pp. 1012–1013]. We consider Chernoff’s problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed. We obtain an integral equation which characterizes the optimal decision rule and find its solution numerically.

We present an analytical description of the class of unsteady vortex surface waves generated by non- uniformly distributed, time-harmonic pressure. The fluid motion is described by an exact solution of the equations of hydrodynamics generalizing the Gerstner solution. The trajectories of the fluid particles are circumferences. The particles on a free surface rotate around circumferences of the same radii, with the centers of the circumferences lying on different horizons. A family of waves has been found in which a variable pressure acts on a limited section of the free surface. The law of external pressure distribution includes an arbitrary function. An example of the evolution of a non-uniform wave packet is considered. The wave and pressure profiles, as well as vorticity distribution are studied. It is shown that, in the case of a uniform traveling wave of external pressure, the Gerstner solution is valid but with a different form of the dispersion relation. A possibility of observing the studied waves in laboratory and in the real ocean is discussed.

The cell formation problem (CFP) is an NP-hard optimization problem considered for cell manufacturing systems. Because of its high computational complexity several heuristics have been developed for solving this problem. In this paper we present a branch and bound algorithm which provides exact solutions of the CFP. This algorithm finds optimal solutions for 13 problems of the 35 popular benchmark instances from the literature.

**Long-term deep bed filtration in porous media with size exclusion particle capture mechanism** **is studied.**** For mono dispersed suspension and transport in porous media whit distributed pore sizes, the micro stochastic model allows for upscaling and the exact solution is derived for the obtained macro scale equation system.** **Results show that** **transient pore size distribution and nonlinear relation between the filtration coefficient and captured particle concentration during suspension filtration and retention are the main features of long-term deep bed filtration, which generalises the classical deep bed filtration model and its latter modifications. Furthermore, the exact solution demonstrates earlier breakthrough and lower breakthrough concentration for larger particles. Among all the pores with different sizes, the ones with intermediate sizes (between the minimum pore size and the particle size) vanish first. Total concentration of all the pores smaller than the particles** **turns to zero asymptotically when time tends to infinity, which corresponds to complete pluggi****ng of smaller pores.**

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

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.