A set of recursive relations satisfied by Selberg-type integrals involvingmonomial symmetric polynomials are derived, generalizing previously knownresults. These formulas provide a well-defined algorithm for computingSelberg-Schur integrals whenever the Kostka numbers relating Schur functionsand the corresponding monomial polynomials are explicitly known.

Schroedinger's equation with the attractive potentialV(r) = -Z/(r^q+ b^q)^(1/q), Z > 0, b > 0, q >= 1, is shown, for generalvalues of the parameters Z and b, to be reducible to the confluent Heunequation in the case q=1, and

We consider the elastic scattering in deformed space with minimal length. Wegive the basic relation for the elastic scattering in deformed space. We alsoinvestigate the partial wave method in deformed space. It is shown that therelations for the scattering amplitude

We construct a conformally invariant random family of closed curves in theplane by welding of random homeomorphisms of the unit circle given in terms ofthe exponential of Gaussian Free Field. We conjecture that our curves arelocally related to SLE$(\kappa)$ for

In these notes we introduce the basic concepts regarding quantumintegrability. Special emphasis is given on the algebraic content of integrablemodels. The associated algebras are essentially described by the Yang-Baxterand boundary Yang-Baxter equations depending on the choice of boundaryconditions. The relation

The (M,K)-reduced non-autonomous discrete KP equation is linearised on thePicard group of an algebraic curve. As an application, we construct thetafunction solutions to the initial value problem of some special discrete KPequation.

The averaged learning equation (ALEH) applicable to the principal componentanalyzer is studied from both quantum information geometry and dynamical systemviewpoints. On the quantum information space (QIS), the space of regulardensity matrices endowed with the quantum SLD-Fisher metric, a gradient systemis

For the case of reduction onto the non-zero momentum level, in the problem ofthe path integral quantization of a scalar particle motion on a smooth compactRiemannian manifold with the given free isometric action of the compactsemisimle Lie group, the path

Main objects of uniformization of the curve $y^2=x^5-x$ are studied: itsBurnside's parametrization, corresponding Schwarz's equation, and accessoryparameters. As a result we obtain the first examples of solvable Fuchsianequations on torus and exhibit number-theoretic integer $q$-series foruniformizing functions, relevant modular forms,