Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space L2(Rd). The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group Sp(Rd), which guarantees the square-integrability of the associated metaplectic wave-packet representation on L2(Rd)

Path Integral Approach to Time-Fractional Quantum Mechanics

The Schrödinger equation which is fractional in space only has been previously derived by Laskin in terms of the Riesz fractional derivative, and the familiar Schrödinger equation is recovered when the fractional order equals 2. The objective of the present thesis is to derive a Schrödinger equation which is fractional in time, such that the standard Schrödinger equation is recovered when the fractional order equals unity, using the path integral method of Feynman. This time-fractional Schrödinger equation will be solved for a free particle, and the fractional wave packet and Green\u27s function solutions will be obtained. Other topics such as the uncertainty product of a Gaussian under fractionalized time will be discussed.It will be shown that the action integral itself must be fractionalized to the same order as the Lagrangian used for the Feynman path integral kernel, in order to maintain the correct order of the fractional derivative in the resulting Schrödinger equation. This suggests that all fractional classical mechanics problems involving Hamilton\u27s principle must be treated in this way as well.In order to maintain correct units and the normalization condition for all fractional orders, it is suggested that space and time be fractionalized as a pair, with a related fractal index, suggesting a fundamental relationship between fractal space and fractal time similar to standard spacetime

Combining 2D2D synchrosqueezed wave packet transform with optimization for crystal image analysis

We develop a variational optimization method for crystal analysis in atomic resolution images, which uses information from a 2D synchrosqueezed transform (SST) as input. The synchrosqueezed transform is applied to extract initial information from atomic crystal images: crystal defects, rotations and the gradient of elastic deformation. The deformation gradient estimate is then improved outside the identified defect region via a variational approach, to obtain more robust results agreeing better with the physical constraints. The variational model is optimized by a nonlinear projected conjugate gradient method. Both examples of images from computer simulations and imaging experiments are analyzed, with results demonstrating the effectiveness of the proposed method

Artificial electromagnetism for neutral atoms: Escher staircase and Laughlin liquids

We show how lasers may create fields which couple to neutral atoms in the same way that the electromagnetic fields couple to charged particles. These fields are needed for using neutral atoms as an analog quantum computer for simulating the properties of many-body systems of charged particles. They allow for seemingly paradoxical geometries, such as a ring where atoms continuously reduce their potential energy while moving in a closed path. We propose neutral atom experiments which probe quantum Hall effects and the interplay between magnetic fields and periodic potentials.Comment: 4 pages, 1 color figure, RevTeX 4; v2 Revised introduction, additional reference

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Quantitative tools for seismic stratigraphy and lithology characterization

Seismological images represent maps of the earth's structure. Apparent bandwidth limitation of seismic data prevents successful estimation of transition sharpness by the multiscale wavelet transform. We discuss the application of two recently developed techniques for (non-linear) singularity analysis designed for bandwidth limited data, such as imaged seismic reflectivity. The first method is a generalization of Mallat's modulus maxima approach to a method capable of estimating coarse-grained local scaling/sharpness/Hölder regularity of edges/transitions from data residing at essentially one single scale. The method is based on a non-linear criterion predicting the (dis)appearance of local maxima as a function of the data's fractional integrations/differentiations. The second method is an extension of an atomic decomposition technique based on the greedy Matching Pursuit Algorithm. Instead of the ordinary Spline Wavelet Packet Basis, our method uses multiple Fractional Spline Wavelet Packet Bases, especially designed for seismic reflectivity data. The first method excels in pinpointing the location of the singularities (the stratigraphy). The second method improves the singularity characterization by providing information on the transition's location, magnitude, scale, order and direction (anti-/causal/symmetric). Moreover, the atomic decomposition entails data compression, denoising and deconvolution. The output of both methods produces a map of the earth's singularity structure. These maps can be overlayed with seismic data, thus providing us with a means to more precisely characterize the seismic reflectivity's litho-stratigraphical information content.Massachusetts Institute of Technology. Industry Consorti