Crossed vortex bottle beam trap for single-atom qubits

We demonstrate trapping and quantum state control of single cesium atoms in a 532 nm wavelength bottle beam trap. The three-dimensional trap is formed by crossing two unit charge vortex beams. Single atoms are loaded with 50% probability directly from a magneto-optical trap. We achieve a trapping lifetime of up to 6 s and demonstrate fast Rabi oscillations with a coherence time of T(2)~43±9 ms.     

Finite time ruin probabilities for tempered stable insurance risk processes

We study the probability of ruin before time t$t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae indicate that some care is needed in the choice of parameters to avoid exponential growth (in time) of the ruin probabilities in these models. This, in particular, applies to the inverse Gaussian process when the safety loading is less than one.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Testing for Mean Reversion in Processes of Ornstein-Uhlenbeck Type

In this paper, we analyse processes of Ornstein-Uhlenbeck (OU) type, driven by Lévy processes. This class is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Lévy process with no OU (autoregressive) effect. For an appropriate discretised version of the model, we utilise likelihood methods to test for such a reduction of the OU process to Lévy motion, deriving the distribution of the relevant pseudo-log-likelihood ratio statistics, asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as the length of the observation window grows large. These analyses are non-standard in that the mean reversion parameter vanishes under the null of a pure Lévy process for the data. Despite this we are able to give a very general analysis with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Lévy process. As a special case, for Brownian motion as driving process, we deduce the limiting distribution in a quite explicit way, finding results which generalise the well-known Dickey-Fuller (‘unit-root’) theory. This revised version was published online in June 2006 with corrections to the Cover Date.     

Relative stability and the strong law of large numbers

Summary LetX ₁,X ₂,..., be i.i.d. random variables andS _n=X ₁+X ₂+. +X _n. In this paper we simplify Rogozin's condition forS _n/B _n ±1for someB _n+, which generalises Hinin's condition for relative stability ofS _n. We also consider convergence of subsequences ofS _n/B _n. As an application of our methods, we extend a result of Chow and Robbins to show thatS _n/B _n±1 a.s. for someB _n + if and only if 0<¦EX¦E¦X¦<+ .     

On Complex Formation between 5-Fluorouracil and β-Cyclodextrin in Solution and in the Solid State: IR Markers and Detection of Short-Lived Complexes by Diffusion NMR

In this work, the nuclear magnetic resonance (NMR) and IR spectroscopic markers of the complexation between 5-fluorouracil (5-FU) and β-cyclodextrin (β-CD) in solid state and in aqueous solution are investigated. In the attenuated total reflectance(ATR) spectra of 5-FU/β-CD products obtained by physical mixing, kneading and co-precipitation, we have identified the two most promising marker bands that could be used to detect complex formations: the C=O and C-F stretching bands of 5-FU that experience a blue shift by ca. 8 and 2 cm⁻¹ upon complexation. The aqueous solutions were studied by NMR spectroscopy. As routine NMR spectra did not show any signs of complexation, we have analyzed the diffusion attenuation of spin–echo signals and the dependence of the population factor of slowly diffusing components on the diffusion time (diffusion NMR of pulsed-field gradient (PFG) NMR). The analysis has revealed that, at each moment, ~60% of 5-FU molecules form a complex with β-CD and its lifetime is ca. 13.5 ms. It is likely to be an inclusion complex, judging from the independence of the diffusion coefficient of β-CD on complexation. The obtained results could be important for future attempts of finding better methods of targeted anticancer drug delivery.     

p-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ? _p , the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.     

Purification and Partial Characterization of an Exo-polygalacturonase from Paecilomyces variotii Liquid Cultures

An extracellular polygalacturonase (PG) produced from Paecilomyces variotii was purified to homogeneity through two chromatography steps using DEAE-Fractogel and Sephadex G-100. The molecular weight of P. variotii PG was 77,300 Da by gel filtration and SDS-PAGE. PG had isoelectric point of 4.37 and optimum pH 4.0. PG was very stable from pH 3.0 to 6.0. The extent of hydrolysis of different pectins by the purified enzyme was decreased with an increase in the degree of esterification. PG had no activity toward non-pectic polysaccharides. The apparent K _m and V _max values for hydrolyzing sodium polypectate were 1.84 mg/mL and 432 μmol/min/mg, respectively. PG was found to have temperature optimum at 65 °C and was totally stable at 45 °C for 90 min. Half-life at 55 °C was 50.6 min. Almost all the examined metal cations showed partial inhibitory effects under enzymatic activity, except for Na⁺¹, K⁺¹, and Co⁺² (1 mM) and Cu⁺² (1 and 10 mM).