In the paper the flow in a thin tubular structure is considered. The velocityof the flow stands for a coefficient in the diffusion-convection equation setin the thin structure. An asymptotic expansion of solution is constructed. Thisexpansion is used further for

It is proved that small periodic singular perturbation of a cylindricalwaveguide surface may open a gap in the continuous spectrum of the Dirichletproblem for the Laplace operator. If the perturbation period is long and thecaverns in the cylinder are small,

We prove global existence of the multivariate viscous Burgers equation systemdefined on the whole space or on a domain isomorphic to the $n$-torus and withtime horizon up to infinity and $C^{\infty}$- data (satisfying some growthconditions if the problem is posed

We survey geometric properties which imply the stochastic incompleteness ofthe minimal diffusion process associated to the Laplacian on manifolds andgraphs. In particular, we completely characterize stochastic incompleteness forspherically symmetric graphs and show that, in contrast to the case ofRiemannian manifolds,

The generalized coherent states attached to the Jacobi group realize thesqueezed states. Imposing hermitian conjugacy to the generators of the Jacobialgebra, we find out the form of the weight function appearing in the scalarproduct. We show effectively the orthonormality of

We consider a nonlinear 4th-order degenerate parabolic partial differentialequation that arises in modelling the dynamics of an incompressible thin liquidfilm on the outer surface of a rotating horizontal cylinder in the presence ofgravity. The parameters involved determine a rich variety

We study the Hilbert expansion for small Knudsen number $\varepsilon$ for theVlasov-Boltzmann-Poisson system for an electron gas. The zeroth order termtakes the form of local Maxwellian: $ Our main result states that if theHilbert expansion is valid at $t=0$ for

For models of non-interacting fermions moving within sites arranged on asurface in three dimensional space, there can be obstructions to findinglocalized Wannier functions. We show that such obstructions are $K$-theoreticobstructions to approximating almost commuting, complex-valued matrices bycommuting matrices, and we

The current associated with field emission is greatly dependent on theelectric field at the emitting electrode. This field is a combination of theelectric field in vacuum and the space charge created by the current. Thelatter becomes more important as the

In this paper we generalize the Sato theory to the extended bigraded Todahierarchy (EBTH). We revise the definition of the Lax eqution, give the Satoequations, wave operators and show the existence of $tau$ function $\tau(t)$.Meanwhile we prove the validity of

For cosmological topologically massive gravity at the chiral point wecalculate momentum space 2- and 3-point correlators of operators in thepostulated dual CFT on the cylinder. These operators are sourced by the bulkand boundary gravitons. Our correlators are fully consistent with