We compute the generating functions of a O(n) model (loop gas model) on arandom lattice of any topology. On the disc and the cylinder, they were alreadyknown, and here we compute all the other topologies. We find that thegenerating functions

Let F1 and F2 be independent copies of correlated fractal percolation, withHausdorff dimensions dimH(F1) and dimH(F2). Consider the following question:does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 willcontain an interval? The well known Palis conjecture states that `generically'this should be

The study of integrability of the mathematical physics equations showed thatthe differential equations describing real processes are not integrable withoutadditional conditions. This follows from the functional relation that isderived from these equations. Such a relation connects the differential ofstate functional

A new method to study a stopped hull of SLE(kappa,rho) is presented. In thisapproach, the law of the conformal map associated to the hull is invariantunder a SLE induced flow. The full trace of a chordal SLE(kappa) can be studiedusing

Confining a quantum particle in a compact subinterval of the real line withDirichlet boundary conditions, we identify the connection of theone-dimensional fractional Schr\"odinger operator with the truncated Toeplitzmatrices. We determine the asymptotic behaviour of the product of eigenvaluesfor the $\alpha$-stable

A random phase property establishing a link between quasi-one-dimensionalrandom Schroedinger operators and full random matrix theory is advocated.Briefly summarized it states that the random transfer matrices placed into anormal system of coordinates act on the isotropic frames and lead to

This paper provides a classification result for gravitational instantons withcubic volume growth and cyclic fundamental group at infinity. It proves that acomplete hyperk\"ahler manifold asymptotic to a circle fibration over theEuclidean three-space is either the standard $\rl^3 \times \sph^1$ or

The many-body wave scattering problem is studied in the case where the bodiesare small, $ka \ll 1$, where $a$ is the characteristic size of a body. Thelimiting case when $a \to 0$ and the total number of the small bodies

Let $q(x)$ be real-valued compactly supported sufficiently smooth function,$q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that thescattering data $A(-\beta,\beta,k)$ $\forall \beta\in S^2$, $\forall k>0$determine $q$ uniquely. here $A(\beta,\alpha,k)$ is the scattering amplitude,corresponding to the potential $q$.

We discuss the solution of a nonlinear ordinary differential equation thatappears in a model for MHD viscous flow caused by a shrinking sheet. We proposean accurate numerical solution and derive simple analytical expressions. Ourresults suggest that a recent perturbation treatment

We prove the Bouchard-Mari\~no Conjecture for the framed one-leggedtopological vertex by deriving the Eynard-Orantin type recursion relations fromthe cut-and-join equation satisfied by the relevant triple Hodge integrals.This establishes a version of local mirror symmetry for the local $C^3$geometry with one

We obtain an exact many-body scattering eigenstate in an open quantum dotsystem. The scattering state is not in the form of the Bethe eigenstate in thesense that the wave-number set of the incoming plane wave is not conservedduring the scattering