On the Spectral Properties of Matrices Associated with Trend Filters

This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the latent components of any time series that the filter smooths through the corresponding eigenvalues. A difficulty arises because matrices associated with trend filters are finite approximations of Toeplitz operators and therefore very little is known about their eigenstructure, which also depends on the boundary conditions or, equivalently, on the filters for trend estimation at the end of the sample. Assuming reflecting boundary conditions, we derive a time series decomposition in terms of periodic latent components and corresponding smoothing eigenvalues. This decomposition depends on the local polynomial regression estimator chosen for the interior. Otherwise, the eigenvalue distribution is derived with an approximation measured by the size of the perturbation that different boundary conditions apport to the eigenvalues of matrices belonging to algebras with known spectral properties, such as the Circulant or the Cosine. The analytical form of the eigenvectors is then derived with an approximation that involves the extremes only. A further topic investigated in the paper concerns a strategy for a filter design in the time domain. Based on cut-off eigenvalues, new estimators are derived, that are less variable and almost equally biased as the original estimator, based on all the eigenvalues. Empirical examples illustrate the effectiveness of the method

Tilings and the aztec diamond theorem

Tilings over the plane are analysed in this work, making a special focus on the Aztec Diamond Theorem. A review of the most relevant results about monohedral tilings is made to continue later by introducing domino tilings over subsets of R2. Based on previous work made by other mathematicians, a proof of the Aztec Diamond Theorem is presented in full detail by completing the description of a bijection that was not made explicit in the original work

Spectral decompositions and \LL^2-operator norms of toy hypocoercive semi-groups

56 pagesInternational audienceFor any $a>0$, consider the hypocoercive generators $y\partial_x+a\partial_y^2-y\partial_y$ and $y\partial_x-ax\partial_y+\partial_y^2-y\partial_y$, respectively for (x,y)\in\RR/(2\pi\ZZ)\times\RR and (x,y)\in\RR\times\RR. The goal of the paper is to obtain exactly the \LL^2(\mu_a)-operator norms of the corresponding Markov semi-group at any time, where $\mu_a$ is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models

Engineering optical nonlinearities in metal nanoparticle arrays

Thesis (Ph.D.)--Boston UniversityMetal nanostructures supporting localized surface plasmon (LSP) resonances are an emerging technology for sensing, optical switching, radiative engineering, and solar energy harvesting, among other applications. The unique property of LSP resonances that enable these technologies is their ability to localize and enhance the optical field near the surface of metal nanoparticles. However, many questions still remain regarding the effects of nanoparticle coupling on the linear and nonlinear optical properties of these structures. In this thesis, I investigate the role of long-range photonic and near-field plasmonic coupling on the linear and nonlinear optical properties of metal nanoparticles in periodic and deterministic aperiodic arrays within a combined experimental and theoretical framework. In particular, I have developed optical characterization techniques to study various properties of planar metal nano-cylinder arrays fabricated by electron beam lithography (EBL). These include the effect of Fano-type coupling between structural grating modes and LSP resonances on linear diffraction and second harmonic generation (SHG), the influence of near-field coupling on the efficiency of plasmon enhanced metal photoluminescence (PL), the dependence of two-photon PL (TPPL) on nanoparticle size, and the multi-polar nature of SHG from planar plasmonic arrays. Experimental results are fully supported by linear scattering theory of the near and far-field properties of particle arrays based on a range of analytical, semi-analytical, and fully numerical techniques. The breadth of computational methods used allows the investigation of a wide range of structures including large aperiodic arrays with hundreds of discrete particles and periodic arrays with realistic particle shapes, substrates, and excitation conditions. The technological potential of engineered plasmonic structures is demonstrated by enhanced vibrational sum frequency generation (VSFG) spectroscopy, a novel nonlinear sensing technique. These studies have revealed design principles for engineering the interplay of photonic and plasmonic coupling for future linear and nonlinear plasmonic devices for sensing, switching, and modulation. The optical characterization techniques developed in this thesis may additionally be used across a wide range of devices in photonics and nano-optics

Introduction to Louis Michel’s lattice geometry through group action

Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the central subject of the book. Di¬fferent basic mathematical tools currently used for the description of lattice geometry are introduced and illustrated through applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional lattices and to lattices associated with integrable dynamical systems. Starting from general Delone sets the authors turn to di¬fferent symmetry and topological classifications including explicit construction of orbifolds for two- and three-dimensional point and space groups

Defects of micropolar continua in Riemann-Cartan manifolds and its applications

We derive equations of motion and its solutions in the form of solitons from deformational energy functionals of a coupled system of microscopic and macroscopic deformations. Then criteria in constructing the chiral energy functional is specified to be included to obtain soliton-like solutions. We show various deformational measures, used in deriving the soliton solutions, can be written when both curvature and torsion are allowed, especially by means of microrotations and its derivatives. Classical compatibility conditions are re-interpreted leading to a universal process to derive a distinct set of compatibility conditions signifying a geometrical role of the Einstein tensor in Riemann-Cartan manifolds. Then we consider position-dependent axial configurations of the microrotations to construct intrinsically conserved currents. We show that associated charges can be written as integers under a finite energy requirement in connection with homotopic considerations. This further leads to a notion of topologically stable defects determined by invariant winding numbers for a given solution classification. Nematic liquid crystals are identified as a projective plane from a sphere hinted by the discrete symmetry in its directors. Order parameters are carefully defined to be used both in homotopic considerations and free energy expansion in the language of microcontinua. Micropolar continua are shown to be the general case of nematic liquid crystals in projective geometry, and in formulations of the order parameter, which is also the generalisation of the Higgs isovectors. Lastly we show that defect measures of pion fields description of the Skyrmions are related to the defect measures of the micropolar continua via correspondences between its underlying symmetries and compatibility conditions of vanishing curvature