The results of an analysis of turbulent pipe flow based on a Karhunen-Loève decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Reτ = 150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categor...

Any closed, connected Riemannian manifold M can be smoothly embedded by its Laplacian eigenfunction maps into Rm for some m. We call the smallest such m the maximal embedding dimension of M. We show that the maximal embedding dimension of M is bounded from above by a constant depending only on the dimension of M, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a vol...

We consider the p-Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite p and investigate the limit problem as p →∞.

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p > 1, we investigate the limit problem as p → 1.

In this paper we construct a version of Ricci flow for noncommutative 2-tori, based on a spectral formulation in terms of the eigenvalues and eigenfunction of the Laplacian and recent results on the Gauss–Bonnet theorem for noncommutative tori.

The biorthogonality condition for Stokes flow in annular trenches bounded by horizontal parallel planes and concentric vertical cylinders is derived. This condition, is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding Stokes flow problem.

With the help of squared eigenfunction potential, action Virasoro symmetry on tau function constrained discrete KP hierarchy is derived.

A class of boundary value problems, that has application in the propagation of waves along ducts in which the boundaries are wave-bearing, is considered. This class of problems is characterised by the presence of high order derivatives of the dependent variable(s) in the duct boundary conditions. It is demonstrated that the underlying eigenfunctions are linearly dependent and, most significantl...

In previous work we introduced and studied a function R(a+, a−, c; v, v̂) that is a generalization of the hypergeometric function 2F1 and the Askey–Wilson polynomials. When the coupling vector c ∈ C is specialized to (b, 0, 0, 0), b ∈ C, we obtain a function R(a+, a−, b; v, 2v̂) that generalizes the conical function specialization of 2F1 and the q-Gegenbauer polynomials. The function R is the joi...

We consider the following eigenvalue problem: −Δu f u λu, u u x , x ∈ B {x ∈ R3 : |x| < 1}, u 0 p > 0, u||x| 1 0, where p is an arbitrary fixed parameter and f is an odd smooth function. First, we prove that for each integer n ≥ 0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r |x| ∈ 0, 1 . For p > 0 sufficiently small, such...