This paper is concerned with weak solution of a mixed boundary value problem for the biharmonic equation in the plane. Using Green’s formula, the problem is converted into a system of Fredholm integral equations for the unknown data on different part of the boundary. Existence and uniqueness of the solutions of the system of boundary integral equations are established in appropriate Sobolev spa...

In accordance with the problem of the need for calculation formula of the straight wall flume flow in open channel by experiment, the method of thee boundary layer is proposed; Analysis with boundary layer displacement thickness is studied, and the energy equation, continuous equation and the theory and critical depth of flume flow coefficient, velocity coefficient is deduced using the boundary...

We prove an analog of Krein’s resolvent formula expressing the resolvents of self-adjoint extensions in terms of boundary conditions. Applications to quantum graphs and systems with point interactions are discussed. AMS classification scheme numbers: 46N50, 47A06, 47A10 PACS numbers: 02.30.Tb, 02.60.Lj Krein’s resolvent formula [1] is a powerful tool in the spectral analysis of self-adjoint ext...

We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of M0n. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology ...

Given a finite rhombus tiling of polygonal region in the plane, associated critical Z-invariant Ising model is invariant under star-triangle transformations. We give simple matrix formula describing spin correlations between boundary vertices terms shape region. When regular polygon, our becomes an explicit trigonometric sum.