Decompositions of Generalized Wavelet Representations

Let $N$ be a simply connected, connected nilpotent Lie group which admits a uniform subgroup $\Gamma.$ Let $\alpha$ be an automorphism of $N$ defined by $\alpha\left( \exp X\right) =\exp AX.$ We assume that the linear action of $A$ is diagonalizable and we do not assume that $N$ is commutative. Let $W$ be a unitary wavelet representation of the semi-direct product group $\left\langle \cup_{j\in\mathbb{Z}}\alpha^{j}\left( \Gamma\right) \right\rangle \rtimes\left\langle \alpha\right\rangle$ defined by $W\left( \gamma,1\right) =f\left( \gamma^{-1}x\right)$ and $W\left( 1,\alpha\right) =\left\vert \det A\right\vert ^{1/2}f\left( \alpha x\right) .$ We obtain a decomposition of $W$ into a direct integral of unitary representations. Moreover, we provide an explicit unitary operator intertwining the representations, a precise description of the representations occurring, the measure used in the direct integral decomposition and the support of the measure. We also study the irreducibility of the fiber representations occurring in the direct integral decomposition in various settings. We prove that in the case where $A$ is an expansive automorphism then the decomposition of $W$ is in fact a direct integral of unitary irreducible representations each occurring with infinite multiplicities if and only if $N$ is non-commutative. This work naturally extends results obtained by H. Lim, J. Packer and K. Taylor who obtained a direct integral decomposition of $W$ in the case where $N$ is commutative and the matrix $A$ is expansive, i.e. all eigenvalues have absolute values larger than one

Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets

Over the last few years, needlets have a emerged as a useful tool for the analysis of Cosmic Microwave Background (CMB) data. Our aim in this paper is first to introduce in the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller and Mayeli (2009a,b). We then proceed with an extensive study of the properties of both standard and Mexican needlets; these properties depend on some parameters which can be tuned in order to optimize the performance for a given application. Our second aim in this paper is then to give practical advice on how to adjust these parameters in order to achieve the best properties for a given problem in CMB data analysis. In particular we investigate localization properties in real and harmonic spaces and propose a recipe on how to quantify the influence of galactic and point source masks on the needlet coefficients. We also show that for certain parameter values, the Mexican needlets provide a close approximation to the Spherical Mexican Hat Wavelets (whence their name), with some advantages concerning their numerical implementation and the derivation of their statistical properties.Comment: 40 pages, 11 figures, published version, main modification: added section on more realistic galactic and point source mask

Adaptive Density Estimation on the Circle by Nearly-Tight Frames

This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the $L^{2}$-risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.Comment: 30 pages, 3 figure

Continuous Wavelets on Compact Manifolds

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say 0 \neq f \in \mathcal{S}(\RR^+), and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on \RR^n. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on \RR^n in \mathcal{S}(\RR^n). In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus \TT^2 or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small

Remoción de carga orgánica en lixiviados por medio de un biofiltro empacado con residuos estabilizados

Los sitios de disposición final de residuos sólidos mal operados, causan afectaciones en su entorno y generan problemas de salud pública; estos sitios en general, son concebidos como pasivos ambientales. En el presente estudio se extrajeron residuos sólidos con edad superior a 8 años de la zona clausurada del relleno sanitario de la ciudad de Tuxtla Gutiérrez, Chiapas, México. Los residuos se caracterizaron con los parámetros de humedad, sólidos totales y sólidos volátiles, encontrando una alta estabilidad biológica en los mismos. Posteriormente, con el objetivo de evaluar el potencial biológico en el tratamiento de lixiviados, estos materiales fueron utilizados como lecho de empaque dentro de un biofiltro semi-aeróbico. Durante los ocho meses de monitoreo, el biofiltro registró eficiencias de remoción en DQO entre 60 y 90%, y alrededor de 60% en color, con cargas hidráulicas del orden de los 10-11 L/m3-d. Estos resultados representan de las primeras investigaciones en México usando como material de empaque residuos estabilizados, demostrando con ello, que los biofiltros pueden ser utilizados como una alternativa atractiva para el pretratamiento de lixiviados de rellenos sanitarios

EEG Microstates Temporal Dynamics Differentiate Individuals with Mood and Anxiety Disorders From Healthy Subjects

Electroencephalography (EEG) measures the brain’s electrophysiological spatio-temporal activities with high temporal resolution. Multichannel and broadband analysis of EEG signals is referred to as EEG microstates (EEG-ms) and can characterize such dynamic neuronal activity. EEG-ms have gained much attention due to the increasing evidence of their association with mental activities and large-scale brain networks identified by functional magnetic resonance imaging (fMRI). Spatially independent EEG-ms are quasi-stationary topographies (e.g., stable, lasting a few dozen milliseconds) typically classified into four canonical classes (microstates A through D). They can be identified by clustering EEG signals around EEG global field power (GFP) maxima points. We examined the EEG-ms properties and the dynamics of cohorts of mood and anxiety (MA) disorders subjects (n = 61) and healthy controls (HCs; n = 52). In both groups, we found four distinct classes of EEG-ms (A through D), which did not differ among cohorts. This suggests a lack of significant structural cortical abnormalities among cohorts, which would otherwise affect the EEG-ms topographies. However, both cohorts’ brain network dynamics significantly varied, as reflected in EEG-ms properties. Compared to HC, the MA cohort features a lower transition probability between EEG-ms B and D and higher transition probability from A to D and from B to C, with a trend towards significance in the average duration of microstate C. Furthermore, we harnessed a recently introduced theoretical approach to analyze the temporal dependencies in EEG-ms. The results revealed that the transition matrices of MA group exhibit higher symmetrical and stationarity properties as compared to HC ones. In addition, we found an elevation in the temporal dependencies among microstates, especially in microstate B for the MA group. The determined alteration in EEG-ms temporal dependencies among the cohorts suggests that brain abnormalities in mood and anxiety disorders reflect aberrant neural dynamics and a temporal dwelling among ceratin brain states (i.e., mood and anxiety disorders subjects have a less dynamicity in switching between different brain states)