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50
MODELING OF WAVE RESONANCES IN LOWCONTRAST PHOTONIC CRYSTALS ∗
"... Abstract. Coupledmode equations are derived from Maxwell equations for modeling of lowcontrast cubiclattice photonic crystals in three spatial dimensions. Coupledmode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundaryvalue pr ..."
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Abstract. Coupledmode equations are derived from Maxwell equations for modeling of lowcontrast cubiclattice photonic crystals in three spatial dimensions. Coupledmode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundaryvalue problem for stationary transmission of four counterpropagating and two oblique waves on the plane. Wellposedness of the boundaryvalue 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.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general 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 ..."
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We propose a new method to analyze and represent data recorded on a domain of general 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 diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève 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 application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
Distance function wavelets  Part II: Extended results and conjectures
 CoRR preprint, http://xxx.lanl.gov/abs/cs.CE/0205063, Research report of Simula Research Laboratory
, 2002
"... 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 c ..."
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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
Distance function wavelets  Part I: Helmholtz and convectiondiffusion transforms and series
 CoRR preprint, http://xxx.lanl.gov/abs/cs.CE/0205019, Research report of Simula Research Laboratory
, 2002
"... 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 convectiondiffusion equations, which include the isotropic Helmholtz Fourier (HF) transform and series, the Helmhol ..."
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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 convectiondiffusion equations, which include the isotropic Helmholtz Fourier (HF) transform and series, the HelmholtzLaplace (HL) transform, and the anisotropic convectiondiffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the IIF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the IIF 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
Distance function wavelets  Part III: "Exotic" transforms and series
, 2002
"... 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 ..."
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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. Followup research is still under way. The paper is intended to inspire more research into this promising area
A (2006) Manifold learning and representations for image analysis and visualization
, 2006
"... 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 manifoldvalued. In this thesis some recently proposed spectral methods for manifold learning are applied to a vi ..."
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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 manifoldvalued. In this thesis some recently proposed spectral methods for manifold learning are applied to a visualization problem in medical imaging. 3D 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 realworld application of manifold learning, the contributions to a generic framework for processing of manifoldvalued 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 manifoldvalued signal processing in SO(3), Q, which is useful
On convergence rates equivalency and sampling strategies in in functional deconvolution models,” The Annals of Statistics
, 2010
"... Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. Fo ..."
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Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these
Analytical Solution of Transient Scalar Wave and Diffusion Problems of Arbitrary Dimensionality and Geometry by RBF Wavelet Series
, 2001
"... 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 eigen ..."
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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 timedependent 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
COMPARISON OF THE CLASSICAL BMO WITH THE BMO SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS
, 2006
"... 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 est ..."
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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.