We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(*D* 8) *τ* function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the *τ* functions of the algebraic solutions of the Painlevé III(*D* 8) from the special representations of the Virasoro algebra of the highest weight (*n* + 1/4)2.

We consider expansions of certain multiple integrals and BKP tau functions in characters of orhtogonal and symplectic groups. In particular, we consider character expansions of integrals over orthogonal and over symplectic matrices

We discuss quantum dynamical elliptic R-matrices related to arbitrary complex simple Lie group G. They generalize the known vertex and dynamical R-matrices and play an intermediate role between these two types. The R-matrices are defined by the corresponding characteristic classes describing the underlying vector bundles. The latter are related to elements of the center of G. While the known dynamical R-matrices are related to the bundles with trivial characteristic classes, the Baxter-Belavin-Drinfeld-Sklyanin vertex R-matrix corresponds to the generator of the center Z N of SL(N). We construct the R-matrices related to SL(N)-bundles with an arbitrary characteristic class explicitly and discuss the corresponding IRF models.

Laser excited small metallic clusters are simulated using classical pseudo potential molecular dynamics simulations. Time-dependent distribution functions are obtained from the electron and ion trajectories in order to investigate plasma properties. The question of local thermodynamic equilibrium is addressed, and size effects are considered. Results for the electron distribution in phase space are given, which are interpreted within equilibrium statistical physics. Momentum autocorrelation functions were calculated for different cluster sizes and for different expansion states from the expanding system after the laser–cluster interaction. A resonance behaviour of the autocorrelation function in finite systems was observed. First, results concerning collision frequencies in small clusters are given.

We show that the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) admits a suggestive reformulation through elliptic functions. We also consider one-variable reductions of the dispersionless DKP hierarchy and show that they are described by an elliptic version of the L¨owner equation. With a particular choice of the driving function, the latter appears to be closely related to the Painlev´e VI equation with special choice of parameters.

The two-point correlation tensor of small-scale fluctuations of magnetic field B in a two-dimensional chaotic flow is studied. The analytic approach is developed in the framework of the Kraichnan–Kazantsev model. It is shown that the growth of the field fluctuations takes place in an essentially resistive regime and stops at large times in accordance with the so-called anti-dynamo theorems. The value of B2 is enhanced in the course of the evolution by the magnetic Prandtl number.

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of gl(N). Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce the symmetrized version of the Drinfeld associator by dropping the odd terms. The symmetrized associator gives the same knot invariants, but has a simpler structure and is fully characterized by one symmetric function which we call the Drinfeld prepotential.

We find all formal solutions to the -dependent KP hierarchy. They are characterized by certain Cauchy-like data. The solutions are found in the form of formal series for the tau-function of the hierarchy and for its logarithm (the *F*-function). An explicit combinatorial description of the coefficients of the series is provided.

We study U(1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions reduce to form factors of the exponential fields of the sine-Gordon model at the free-fermion point.

Bounds, expressed in terms of *d* and *N* on *full Bell locality* of a quantum state for *N>*2 nonlocally entangled qudits (of a dimension *d≥2*) mixed with white noise are known, to our knowledge, only within full separability of this noisy *N*-qudit state. For the maximal violation of general Bell inequalities by an *N*-partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in *d,N* , valid for all *d≥*2 and all *N≥*2, on *full Bell locality* under generalized quantum measurements of (i) the *N*-qudit GHZ state mixed with white noise and (ii) an arbitrary *N*-qudit state mixed with white noise. The new full Bell locality bounds are beyond the known ranges for full separability of these noisy *N*-qudit states.