Abstract. Coupled-mode equations are derived from Maxwell equations for modeling of lowcontrast cubic-lattice photonic crystals in three spatial dimensions. Coupled-mode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundary-value problem for stationary transmission of four counter-propagating and two oblique waves on the plane. Well-posedness of the boundary-value problem is proved by using the method of separation of variables and generalized Fourier series. For applications in photonic optics, we compute integral invariants for transmission, reflection, and diffraction of resonant waves.

We propose a new method to analyze and represent data recorded on a domain of gen-eral shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonal-ize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neu-mann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the compu-tation. We also show that our method is better suited for small sample data than the Karhunen-Loève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further ap-plication, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain. Key words: Laplacian eigenfunctions, boundary conditions, Green’s function, spectral decomposition, Karhunen-Loève transform, principal component analysis, heat equation

This report is the second in series [1,2] about my latest advances on the distance function wavelets (DFW). Unlike the common distance functions, e.g., MQ and TPS, which have no provision for scaling and carry out a multiresolution hierarchy by simply dropping or adding some points [3], the DFW is comprised of both the scale and translation arguments. To better understand what will be presented here, the readers are advised to have a look at Report I [1] beforehand. The report is featured with lots of grand conjectures, where firm mathematical underpinnings are conspicuously lacking in most cases. But nevertheless the author assumes that many results have certain physical grounds and are in agreement with the faith that God rules the world with simplicity and beauty

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz- Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the I-IF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the I-IF and HL transforms, a new dimension- dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge

This paper also briefly discusses and conjectures the DFW correspondences of a variety of coordinate variable transforms and series. Practically important, the anisotropic and inhomogeneous DFW's are developed by using the geodesic distance variable. The DFW and the related basis functions are also used in making the kernel distance sigrnoidal functions, which are potentially useful in the artificial neural network and machine learning. As or even worse than the preceding two reports, this study scarifies mathematical rigor and in mm unfetter imagination. Most results are intuitively obtained without rigorous analysis. Follow-up research is still under way. The paper is intended to inspire more research into this promising area

i A manifold is a mathematical concept which generalizes surfaces to higher dimensions. The values of signals and data are sometimes naturally described as points in manifolds – they are manifold-valued. In this thesis some recently proposed spectral methods for manifold learning are applied to a visualization problem in medical imaging. 3-D volume data of the human brain, acquired using Diffusion Tensor MRI, is post processed in a novel way in order to represent and visualize the shape and connectivity of white matter fiber bundles. In addition to this real-world application of manifold learning, the contributions to a generic framework for processing of manifold-valued signals and data consist of the following. 1) The idea of the diffusion mean, which is a preliminary result related to the extrinsic and intrinsic means in certain manifolds. 2) A representation for extrinsic manifold-valued signal processing in SO(3), Q, which is useful

Abstract not found

Contents 1. Fourier series, a historic retrospect .................................................. 1 2. Radial basis function and wavelets .................................................. 3 3. Analytical solution of transient wave problem with RBF wavelet series ...... 7 3.1. Helmholtz eigenvalues and eigenfunctions with RBF ............................ 8 3.2. Analytical solution with RBF wavelet series ..................................... 10 4. Applications to inhomogeneous problems ......................................... 13 5. Extension to other time-dependent equations ..................................... 14 6. Generalized RBF wavelet series and transforms ................................. 14 7. Promises and open problems ........................................................ 16 References ................................................................................. 18 1. Fourier series, a historic retrospect Many of the important concepts of analysis and c

Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space. 1.

The ability to study the interior of an object without destroying it is an important industrial tool. One method of recent interest is thermal imaging. The idea is to use heat energy as a kind of “x-ray”, to form an image of the