The series representation consisting of eigenfunctions as the orthogonal basis is called the Karhunen–Loeve expansion. This paper demonstrates that the determination of eigensolutions using a wavelet-Galerkin scheme for Karhunen–Loeve expansion is computationally equivalent to using wavelet directly for stochastic expansion and simulating the correlated random coefficients using eigen decomposi...

Abstract— A high-accuracy economical iterative method is proposed for calculating the potential and strength of electric field in a three-dimensional inhomogeneous spatially periodic dielectric placed an initially uniform field. The idea underlying algorithm that represented as sum linear function correction, which can be expressed expansion eigenfunctions Laplace operator satisfy appropriate p...

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier t...

The sparse regression framework has been widely used in medical image processing and analysis. However, it has been rarely used in anatomical studies. We present a sparse shape modeling framework using the Laplace-Beltrami (LB) eigenfunctions of the underlying shape and show its improvement of statistical power. Traditionally, the LB-eigenfunctions are used as a basis for intrinsically represen...

An expansion method was used to write a MATHEMATICA program to compute the energy levels and eigenfunctions of a 2-D quantum billiard system with arbitrary shape and dirichlet boundary conditions. One integrable system, the full circle, and one non-integrable system, the stadium, were examined. Chaotic properties were sought in nearest-neighbor energy level spacing distributions (NND). It was o...

The rnapshot method is used to analyze a large eddy simulation of axisymmetric jet flow. An ensemble of realizations is collected using a sampling condition that corresponds to the passage of a large scale vortex at a position six diameters downstream from the nozzle. The analysis is performed separately on a variable composed of the primitive flow quantities and the mass fraction of the materi...

The method of empirical eigenfunctions is developed in a general framework. In particular it is shown that the method of snapshots leads to the determination of the empirical eigenfunctions in any number of dimensions in terms of an equivalent one-dimensional problem. The methodology is discussed within the framework of some turbulent simulations and it is shown how this facilitates the flow an...

We calculate the Regge trajectories of the subleading BFKL singularities and eigenfunctions for the running BFKL pomeron in the color dipole representation. We obtain a viable BFKL-Regge expansion of the proton structure function F2p(x,Q ) in terms of several rightmost BFKL singularities. We find large subleading contributions to F2p(x,Q ) in the HERA kinematical region which explains the lack ...

• Introduction 1. Example: rotationally symmetric eigenfunctions on R 2. Example: translation-equivariant eigenfunctions on H 3. Beginning of construction of solutions 4. K(x, t) is bounded 5. End of construction of solutions 6. Asymptotics of solutions 7. Appendix: asymptotic expansions • Bibliography According to [Erdélyi 1956], Thomé [1] found that differential equations with finite rank irr...

We construct a time-dependent scattering theory for Schrödinger operators on a manifold M with asymptotically conic structure. We use the two-space scattering theory formalism, and a reference operator on a space of the form R×∂M , where ∂M is the boundary of M at infinity. We prove the existence and the completeness of the wave operators, and show that our scattering matrix is equivalent to th...