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## On a new application of the path integrals in polymer statistical physics

We propose a new approach based on the path integral formalism to the calculation of the probability distribution functions of quadratic quantities of the Gaussian polymer chain in d-dimensional space, such as the radius of gyration and potential energy in the parabolic well. In both cases we obtain the exact relations for the characteristic function and cumulants. Using the standard steepest-descent method, we evaluate the probability distribution functions in two limiting cases of the large and small values of corresponding variables.

Abstracts of the 9-th International Workshop on Applied Probability

The partition function of the quantum Heisenberg ferromagnet is represented in a closed functional form. The resulting expression is a path integral for two interacting scalar Bose fields with non-polynomial action.

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.

We develop a first-principle equation of state of salt-free polyelectrolyte solution in the limit of infinitely long flexible polymer chains in the framework of a field-theoretical formalism beyond the linear Debye-Hueckel theory and predict a liquid-liquid phase separation induced by a strong correlation attraction. As a reference system, we choose a set of two subsystems—charged macromolecules immersed in a structureless oppositely charged background created by counterions (polymer one component plasma) and counterions immersed in oppositely charged background created by polymer chains (hard-core one component plasma). We calculate the excess free energy of polymer one component plasma in the framework of modified random phase approximation, whereas a contribution of charge densities’ fluctuations of neutralizing backgrounds we evaluate at the level of Gaussian approximation. We show that our theory is in a very good agreement with the results of Monte Carlo andMDsimulations for critical parameters of liquid-liquid phase separation and osmotic pressure in a wide range of monomer concentration above the critical point, respectively.

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.