114 research outputs found
Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space
(Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2)
(locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be
mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski
space through a Cayley transformation, where other kind of (electromagnetic)
wavelets have already been proposed in the literature. We study the unitary
irreducible representations of the conformal group on the Hilbert spaces
L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square
integrable holomorphic functions with scale dimension \lambda and continuous
mass spectrum, prove the isomorphism (equivariance) between both Hilbert
spaces, admissibility and tight-frame conditions, provide reconstruction
formulas and orthonormal basis of homogeneous polynomials and discuss symmetry
properties and the Euclidean limit of the proposed conformal wavelets. For that
purpose, we firstly state and prove a \lambda-extension of Schwinger's Master
Theorem (SMT), which turns out to be a useful mathematical tool for us,
particularly as a generating function for the unitary-representation functions
of the conformal group and for the derivation of the reproducing (Bergman)
kernel of L^2_h(D_4,d\nu_\lambda). SMT is related to MacMahon's Master Theorem
(MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is
also provided for completeness. Convergence conditions are also studied.Comment: LaTeX, 40 pages, three new Sections and six new references added. To
appear in ACH
Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Using coherent-state techniques, we prove a sampling theorem for Majorana's
(holomorphic) functions on the Riemann sphere and we provide an exact
reconstruction formula as a convolution product of samples and a given
reconstruction kernel (a sinc-type function). We also discuss the effect of
over- and under-sampling. Sample points are roots of unity, a fact which allows
explicit inversion formulas for resolution and overlapping kernel operators
through the theory of Circulant Matrices and Rectangular Fourier Matrices. The
case of band-limited functions on the Riemann sphere, with spins up to , is
also considered. The connection with the standard Euler angle picture, in terms
of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App
Splines and Wavelets on Geophysically Relevant Manifolds
Analysis on the unit sphere found many applications in
seismology, weather prediction, astrophysics, signal analysis, crystallography,
computer vision, computerized tomography, neuroscience, and statistics.
In the last two decades, the importance of these and other applications
triggered the development of various tools such as splines and wavelet bases
suitable for the unit spheres , and the
rotation group . Present paper is a summary of some of results of the
author and his collaborators on generalized (average) variational splines and
localized frames (wavelets) on compact Riemannian manifolds. The results are
illustrated by applications to Radon-type transforms on and
.Comment: The final publication is available at http://www.springerlink.co
Reproducing subgroups of . Part I: algebraic classification
We classify the connected Lie subgroups of the symplectic group
whose elements are matrices in block lower triangular form.
The classification is up to conjugation within . Their study
is motivated by the need of a unified approach to continuous 2D signal
analyses, as those provided by wavelets and shearlets.Comment: 26 page
Children's humor types and psychosocial adjustment
Attempting to understand how humor styles relate to psychological adjustment by correlating these two constructs fails to address the emerging understanding that individuals use combinations of humor styles, and that different combinations may be differentially associated with psychosocial adjustment. Indeed humor types have been identified in adult samples (Galloway, 2010; Leist & MĂŒller, 2013). The main aim of the study was to explore whether similar humor types are evident at a younger age and whether these types can be distinguished
in terms of children's psychological and social well-being. Participants were 1234 adolescents (52% female) aged 11â13 years, drawn from six secondary schools in England. Self-reports of humor styles and psychosocial adjustment were collected at two time points, 6 months apart. A cluster analysis was performed using the child humor styles scores at Time 1. Four humor types were identified: âInterpersonal Humoristsâ (high on aggressive and affiliative humor, low on self-defeating and self-enhancing humor), âSelf-Defeatersâ (high self-defeating humor, low on the other three), âHumor Endorsersâ (high on all four humor styles), and âAdaptive Humoristsâ (high on self-enhancing and affiliative humor, but low on aggressive and self-defeating
humor). âSelf-Defeatersâ scored highest in terms of maladjustment across all of the outcomes measured. Our analyses support the presence of distinctive humor types in childhood and indicate that these are related to psychosocial adjustment
The transmission problem on a three-dimensional wedge
We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge
- âŠ