Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

Adaptive homodyne measurement of optical phase

We present an experimental demonstration of the power of feedback in quantum metrology, confirming the predicted [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)]] superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For measurements performed on weak coherent states with no prior knowledge of the signal phase, adaptive homodyne estimation approaches closer to the intrinsic quantum uncertainty than any previous technique. Our results underscore the importance of real-time feedback for reaching quantum limits in measurement and control.     

Ergodicity for the Randomly Forced 2D Navier–Stokes Equations

We study space-periodic 2D Navier–Stokes equations perturbed by an unbounded random kick-force. It is assumed that Fourier coefficients of the kicks are independent random variables all of whose moments are bounded and that the distributions of the first N ₀ coefficients (where N ₀ is a sufficiently large integer) have positive densities against the Lebesgue measure. We treat the equation as a random dynamical system in the space of square integrable divergence-free vector fields. We prove that this dynamical system has a unique stationary measure and study its ergodic properties.     

Osteoporosis detection in postmenopausal women using axial transmission multi-frequency bone ultrasonometer: Clinical findings

The objective of this study was to evaluate if the Bone UltraSonic Scanner (BUSS) can detect osteoporosis in postmenopausal women. BUSS is an axial transmission multi-frequency ultrasonometer for acquisition of wave propagation profiles along the proximal anterior tibia. We derived 10 diagnostically significant BUSS parameters that were then compared with the DXA spine T-score, which was used in this study as the “gold standard” for the assessment of osteoporosis (T-score <−2.5). BUSS wave parameters were studied in 331 postmenopausal women examined by 9 trained operators at 3 clinical sites with use of 3 devices. The efficiency of each BUSS parameter in osteoporosis detection was assessed using a receiver operating characteristic curve analysis. Area under the curve (AUC) for each of 10 parameters ranged from 58.1% to 70.2%. Using these parameters a linear classifier was derived which provided at its output 83.0% AUC, 87.7% sensitivity and 63.2% specificity to DXA-identified osteoporosis. The results of this study confirm BUSS’s capability to detect osteoporosis in postmenopausal women.     

Highly enantioselective direct alkylation of arylacetic acids with chiral lithium amides as traceless auxiliaries

A direct, highly enantioselective alkylation of arylacetic acids via enediolates using a readily available chiral lithium amide as a stereodirecting reagent has been developed. This approach circumvents the traditional attachment and removal of chiral auxiliaries used currently for this type of transformation. The protocol is operationally simple, and the chiral reagent is readily recoverable.     

Exponential Dichotomy and Time-Bounded Solutions for First-Order Hyperbolic Systems

The paper is devoted to studying a class of strongly hyperbolic systems of the first order. We show that if the characteristic roots of the full symbol are outside an open strip containing the real axis, then the homogeneous system possesses an exponential dichotomy and the inhomogeneous system is solvable in the space of time-bounded and almost periodic functions. We also discuss some results on the behavior of solutions for nonlinear equations in the neighborhood of a stationary point.