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## Stochastic processes with $Z_N$ symmetry and complex Virasoro representations. The partition functions.

In a previous letter (Alcaraz F C *et al* 2014 *J. Phys. A: Math. Theor.* **47** 212003) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley–Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a *Z**N* symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For *N* even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.

A problem of minimization of integral losses on given horizon is considered for stochastic system in continuous time. The losses occur in jump times of a Poisson process, and represent continuous convex function of control parameter on convex compact finite-dimensional set. At the jump times an oracle provide stochastically perturbed sub-gradient of the loss function, bounded in mean squares; the noise is additive and centered. Control strategy generated by Mirror Descent algorithm is suggested. For the strategy an explicit upper bound for integral loss discrepancy over its minimum is proved. Example of such strategy application to queueing model is examined.

We consider the AGT correspondence in the context of the conformal field theory M(p, p')\otimes H, where M(p,p') is the minimal model based on the Virasoro algebra labeled by two co-prime integers p,p' and H is the free boson theory based on the Heisenberg algebra. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in M(p, p')\otimes H leads to ill-defined or incorrect expressions.

We propose the procedure to make this expressions are well defined and check these proposal in two cases: 1. 1-point torus, when the operator insertion is the identity, and 2. The 6-point Ising conforma block on the sphere that involves six Ising magnetic operators.

Over the last 50 years in different areas such as decision theory, information processing, and data mining, the interest to extend probability theory and statistics has grown. The common feature of those attempts is to widen frameworks for representing different kinds of uncertainty: randomness, imprecision, vagueness, and ignorance. The scope is to develop more flexible methods to analyze data and extract knowledge from them. The extension of classical methods consists in “softening” them by means of new approaches involving fuzzy set theory, possibility theory, rough sets, or having their origin in probability theory itself, like imprecise probabilities, belief functions, and fuzzy random variables. Data science aims at developing automated methods to analyze massive amounts of data and extract knowledge from them. In the recent years the production of data is dramatically increasing. Every day a huge amount of data coming from everywhere is collected: mobile sensors, sophisticated instruments, transactions, Web logs, and so forth. This trend is expected to accelerate in the near future. Data science employs various programming techniques and methods of data wrangling, data visualization, machine learning, and probability and statistics. The soft methods proposed in this volume represent a suit of tools in these fields that can also be useful for data science. The volume contains 65 selected contributions devoted to the foundation of uncertainty theories such as probability, imprecise probability, possibility theory, soft methods for probability and statistics. Some of them are focused on robustness, non-precise data, dependence models with fuzzy sets, clustering, mathematical models for decision theory and finance.

This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of functionals of diffusions stopped at different random times, the Brownian local time, diffusions with jumps, and an invariance principle for random walks and local times. Supported by carefully selected material, the book showcases a wealth of examples that demonstrate how to solve concrete problems by applying theoretical results. It addresses a broad range of applications, focusing on concrete computational techniques rather than on abstract theory. The content presented here is largely self-contained, making it suitable for researchers and graduate students alike.

We consider the M(2,3) Minimal Liouville gravity, whose states in the gravity sector are represented by irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes. This construction is based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We construct an algebra of operators acting on the BRST cohomology space. The operator algebra of physical states is established by use of these operators.

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.