Physical Sciences and Mathematics Commons^™

Concentration Theorems For Orthonormal Sequences In A Reproducing Kernel Hilbert Space, Travis Alvarez

All Dissertations

Let H be a reproducing kernel Hilbert space with reproducing kernel elements {Kx} indexed by a measure space {X,mu}. If H can be embedded in L2(X,mu), then H can be viewed as a framed Hilbert space. We study concentration of orthonormal sequences in such reproducing kernel Hilbert spaces.

Defining different versions of concentration, we find quantitative upper bounds on the number of orthonormal functions that can be classified by such concentrations. Examples are shown to prove sharpness of the bounds. In the cases that we can add "concentrated" orthonormal vectors indefinitely, the growth rate of doing so is shown.

Sparse Domination Of The Martingale Transform, Michael Scott Kutzler

Mathematics & Statistics ETDs

Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators. The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator. Sparse operators are based on a sparse family which is simply a subset of a dyadic grid. We also show the A2 conjecture for the Martingale Transform which follows from the sparse domination of the Martingale Transform and the A2 conjecture for sparse operators.

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A Brief On Characteristic Functions, Austin G. Vandegriffe

Graduate Student Research & Creative Works

Characteristic functions (CFs) are often used in problems involving convergence in distribution, independence of random variables, infinitely divisible distributions, and stochastics. The most famous use of characteristic functions is in the proof of the Central Limit Theorem, also known as the Fundamental Theorem of Statistics. Though less frequent, CFs have also been used in problems of nonparametric time series analysis and in machine learning. Moreover, CFs uniquely determine their distribution, much like the moment generating functions (MGFs), but the major difference is that CFs always exists, whereas MGFs can fail, e.g. the Cauchy distribution. This makes CFs more robust in …

Survey Of Results On The Schrodinger Operator With Inverse Square Potential, Richardson Saint Bonheur

Electronic Theses and Dissertations

In this paper we present a survey of results on the Schrodinger operator with Inverse ¨ Square potential, L_a= −∆ + a/|x|^2 , a ≥ −( d−2/2 )^2. We briefly discuss the long-time behavior of solutions to the inter-critical focusing NLS with an inverse square potential(proof not provided). Later we present spectral multiplier theorems for the operator. For the case when a ≥ 0, we present the multiplier theorem from Hebisch [12]. The case when 0 > a ≥ −( d−2/2 )^2 was explored in [1], and their proof will be presented for completeness. No improvements on the sharpness …

Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, David Edward Weirich

Mathematics & Statistics ETDs

A so-called space of homogeneous type is a set equipped with a quasi-metric and a doubling measure. We give a survey of results spanning the last few decades concerning the geometric properties of such spaces, culminating in the description of a system of dyadic cubes in this setting whose properties mirror the more familiar dyadic lattices in R^n . We then use these cubes to prove a result pertaining to weighted inequality theory over such spaces. We develop a general method for extending Bellman function type arguments from the real line to spaces of homogeneous type. Finally, we uses this …

Heun Polynomials In The Construction Of Vector Valued Slepian Functions On A Spherical Cap, Thomas Anthony Ventimiglia

Theses and Dissertations

I summarize the existing work on the problem of finding vector valued Slepian functions on the unit sphere: separable vector fields whose energy is concentrated within a compact region; in this case, a spherical cap. The radial and tangential components are independent for an appropriate choice of basis, and for each component the problem is recast as that of finding real eigenfunctions of an integral operator. There exist Sturm-Liouville differential operators that commute with these integral operators and hence share their eigenfunctions. Therefore, the radial and tangential eigenfunctions are solutions to second order linear ODEs. After introducing the Heun differential …

Waveforms For Optimal Sub-Kev High-Order Harmonics With Synthesized Two- Or Three-Colour Laser Fields, Cheng Jin, Guoli Wang, Hui Wei, Anh-Thu Le, C. D. Lin

Physics Faculty Research & Creative Works

High-order harmonics extending to the X-ray region generated in a gas medium by intense lasers offer the potential for providing tabletop broadband light sources but so far are limited by their low conversion efficiency. Here we show that harmonics can be enhanced by one to two orders of magnitude without an increase in the total laser power if the laser's waveform is optimized by synthesizing two- or three-colour fields. The harmonics thus generated are also favourably phase-matched so that radiation is efficiently built up in the gas medium. Our results, combined with the emerging intense high-repetition MHz lasers, promise to …

The Boundedness Of Hausdorff Operators On Function Spaces, Xiaoying Lin

Theses and Dissertations

For a fixed kernel function $\Phi$, the one dimensional Hausdorff operator is defined in the integral form by

\[

\hphi (f)(x)=\int_{0}^{\infty}\frac{\Phi(t)}{t}f(\frac{x}{t})\dt.

\]

By the Minkowski inequality, it is easy to check that the Hausdorff operator is bounded on the Lebesgue spaces $L^{p}$ when $p\geq 1$, with some size condition assumed on the kernel functions $\Phi$. However, people discovered that the above boundedness property is quite different on the Hardy space $H^{p}$ when $0

In this thesis, we first study the boundedness of $\hphi$ on the Hardy space $H^{1}$, and on the local Hardy space $h^{1}(\bbR)$. Our work shows that for …

Winding Resistance And Winding Power Loss Of High-Frequency Power Inductors, Rafal P. Wojda

Browse all Theses and Dissertations

The scope of this research is concentrated on analytical winding size optimization (thickness or diameter) of high-frequency power inductors wound with foil, solid-round wire, multi-strand wire, and litz-wire conductors.

The first part of this research concerns analytical optimization of the winding size (thickness or diameter) for the inductors conducting a sinusoidal current. Estimation of winding resistance in individual inductor layers made of foil, taking into account the skin and proximity effects is performed. Approximated equations for the winding power loss in each layer are given and the optimal values of foil thickness for each layer are derived.

A low- and …

Influence Of Gas Pressure On High-Order-Harmonic Generation Of Ar And Ne, Guoli Wang, Cheng Jin, Anh-Thu Le, C. D. Lin

Physics Faculty Research & Creative Works

We study the effect of gas pressure on the generation of high-order harmonics where harmonics due to individual atoms are calculated using the recently developed quantitative rescattering theory, and the propagation of the laser and harmonics in the medium is calculated by solving the Maxwell's wave equation. We illustrate that the simulated spectra are very sensitive to the laser focusing conditions at high laser intensity and high pressure since the fundamental laser field is severely reshaped during the propagation. By comparing the simulated results with several experiments we show that the pressure dependence can be qualitatively explained. The lack of …

Analysis Of Effects Of Macroscopic Propagation And Multiple Molecular Orbitals On The Minimum In High-Order Harmonic Generation Of Aligned Co₂, Cheng Jin, Anh-Thu Le, C. D. Lin

Physics Faculty Research & Creative Works

We report theoretical calculations of the effect of the multiple-orbital contribution in high-order harmonic generation (HHG) of aligned CO₂ with the inclusion of macroscopic propagation of harmonic fields in the medium. Our results show very good agreement with recent experiments for the dynamics of the minimum in HHG spectra as laser intensity or alignment angle changes. Calculations are carried out to check how the position of the minimum in HHG spectra depends on the degrees of molecular alignment, laser-focusing conditions, and the effects of alignment-dependent ionization rates of the different molecular orbitals. These analyses help to explain why the minima …

Medium Propagation Effects In High-Order Harmonic Generation Of Ar And N₂, Cheng Jin, Anh-Thu Le, C. D. Lin

Physics Faculty Research & Creative Works

We report theoretical calculations of high-order harmonic generation (HHG) by intense infrared lasers in atomic and molecular targets taking into account the macroscopic propagation of both fundamental and harmonic fields. On the examples of Ar and N₂, we demonstrate that these ab initio calculations are capable of accurately reproducing available experimental results with isotropic and aligned target media. We further present detailed analysis of HHG intensity and phase under various experimental conditions, in particular, as the wavelength of the driving laser changes. Most importantly, our results strongly support the factorization of HHG at the macroscopic level into a product of …

Quantitative Rescattering Theory For High-Order Harmonic Generation From Molecules, Anh-Thu Le, R. R. Lucchese, S. Tonzani, Toru Morishita, C. D. Lin

Physics Faculty Research & Creative Works

The quantitative rescattering theory (QRS) for high-order harmonic generation (HHG) by intense laser pulses is presented. According to the QRS, HHG spectra can be expressed as a product of a returning electron wave packet and the photorecombination differential cross section of the laser-free continuum electron back to the initial bound state. We show that the shape of the returning electron wave packet is determined mostly by the laser. The returning electron wave packets can be obtained from the strong-field approximation or from the solution of the time-dependent Schrödinger equation (TDSE) for a reference atom. The validity of the QRS is …

Probing Molecular Frame Photoionization Via Laser Generated High-Order Harmonics From Aligned Molecules, Anh-Thu Le, R. R. Lucchese, M. T. Lee, C. D. Lin

Physics Faculty Research & Creative Works

Present experiments cannot measure molecular frame photoelectron angular distributions (MFPAD) for ionization from the outermost valence orbitals of molecules. We show that the details of MFPAD can be retrieved with high-order harmonics generated by infrared lasers from aligned molecules. Using accurately calculated photoionization transition dipole moments for fixed-in-space molecules, we show that the dependence of the magnitude and phase of the high-order harmonics on the alignment angle of the molecules observed in recent experiments can be quantitatively reproduced. This result provides the needed theoretical basis for ultrafast dynamic chemical imaging using infrared laser pulses.

Retrieving Photorecombination Cross Sections Of Atoms From High-Order Harmonic Spectra, Shinichiro Minemoto, Toshihito Umegaki, Yuichiro Oguchi, Toru Morishita, Anh-Thu Le, Shinichi Watanabe, Hirofumi Sakai

Physics Faculty Research & Creative Works

We observe high-order harmonic spectra generated from a thin atomic medium, Ar, Kr, and Xe, by intense 800-nm and 1300-nm femtosecond pulses. A clear signature of a single-atom response is observed in the harmonic spectra. Especially in the case of Ar, a Cooper minimum, reflecting the electronic structure of the atom, is observed in the harmonic spectra. We successfully extract the photorecombination cross sections of the atoms in the field-free condition with the help of an accurate recolliding electron wave packet. The present protocol paves the way for exploring ultrafast imaging of molecular dynamics with attosecond resolution.