In this paper, we study Bessel processes of dimension δ ≡ 2(1 − μ), with 0 < δ < 2, and some related martingales and random times. Our approach is based on martingale techniques and the general theory of stochastic processes (unlike the usual approach based on excursion theory), although for 0 < δ < 1, these processes are even not semimartingales. The last time before 1 when a Bessel process hi...

Title of dissertation: HOLLOW BESSEL TUNNEL GUIDE: THEORY AND EXPERIMENT Narupon Chattrapiban Doctor of Philosophy, 2012 Dissertation directed by: Professor Wendell T. Hill, III Department of Physics The main objectives of this thesis were to create and investigate properties and applications of all-optical atomic guides, which are potential building blocks for atom optics experiments. Specific...

Diffraction is a cornerstone of optical physics and has implications for the design of all optical systems. The paper discusses the so-called ‘non-diffracting’ light field, commonly known as the Bessel beam. Approximations to such beams can be experimentally realized using a range of different means. The theoretical foundation of these beams is described and then various experiments that make u...

Recently, harmonic imaging has been shown to increase image quality in medical imaging. This is believed to be caused by decreased sidelobes. It is also due to a decreased sensitivity to near-field aberrations owing to the lower transmission frequency, and the lower relative amplitude of the second harmonic in the near-field. One of the problems of limited diffraction beams has been the sidelob...

Title of Dissertation: Two Experiments with Cold Atoms: I. Application of Bessel Beams for Atom Optics, and II. Spectroscopic Measurements of Rydberg Blockade Effect Ilya Arakelyan, Doctor of Philosophy, 2010 Dissertation directed by: Prof. Wendell Hill, III Department of Physics In this dissertation we report the results of two experimental projects with lasercooled rubidium atoms: I. Applicat...

In this letter, we explore the possibilities of generating diffraction-free caustics over a predetermined range. For the first time we demonstrate the possibility to generate diffraction-free caustic beams. Caustic beams are a topic of increased interest over the past few years. Diffraction-free caustics may find applications in situations where large depth of field and uniform axial energy dis...

Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in L(R). The characterizations are applied to Bessel sequences which have an affine struc...

We propose a novel algebraic characterisation of the classical notion of validity in terms of boolean rings, called entailment multipliers. We demonstrate the existence of such multipliers and show how they can be used to derive stronger entailment statements. An interesting property of multipliers lies in their behaviour as invariants in a proof, a fact that is used to show how several inferen...