Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function

The y-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic y-basin of attraction of the equilibrium

General Relativistic Geodetic Spin Precession in Binary Pulsar B1913+16: Mapping the Emission Beam in Two Dimensions

We have carefully measured the pulse profile of the binary pulsar PSR B1913+16 at 21 cm wavelength for twenty years, in order to search for variations that result from general relativistic geodetic precession of the spin axis. The profile width is found to decrease with time in its inner regions, while staying essentially constant on its outer skirts. We fit these data to a model of the beam shape and precession geometry. Four equivalent solutions are found, but evolutionary considerations and polarization data select a single preferred model. While the current data sample only a limited range of latitudes owing to the long precessional cycle, the preferred model shows a beam elongated in the latitude direction and hourglass--shaped.Comment: Accepted by AP

Validation of quantitative measures of rotatory knee laxity

Background: Prior attempts to quantify the pivot-shift examination have been too invasive or impractical for clinical use. A noninvasive method for quantifying rotatory knee laxity is needed. Hypothesis: Greater quantitative measurements of rotatory knee laxity (both of the involved knee as well as compared with the contralateral healthy knee) are associated with an increasing clinical pivot-shift grade. Study Design: Cross-sectional study; Level of evidence, 3. Methods: A total of 103 patients undergoing anatomic single-bundle anterior cruciate ligament (ACL) reconstruction at 4 international centers underwent a standardized pivot-shift test preoperatively on both knees while anesthetized. Clinical grading of the pivot shift was performed according to the International Knee Documentation Committee (IKDC) knee ligament rating system. Two different quantitative technologies were used to measure rotatory knee laxity: an inertial sensor and an image analysis were independently used to measure tibial acceleration and lateral compartment translation, respectively, during the pivot-shift test. Patients were dichotomized to "high-grade" (abnormal and severely abnormal) or "low-grade" (normal and nearly normal) rotatory knee laxity groups based on the clinical pivot-shift test result of the involved side. Tibial acceleration and lateral compartment translation of the involved knee and the side-to-side difference between the involved and contralateral knees were separately compared between the high- and low-grade rotatory knee laxity groups utilizing t tests; significance was set at P <.05. Results: Forty-three patients were in the low-grade rotatory knee laxity group, and 60 patients were in the high-grade rotatory knee laxity group. Patients in the high-grade knee laxity group had significantly higher lateral compartment translation as measured with the image analysis (involved knee: 3.8 ± 2.3 mm; side-to-side difference: 2.5 ± 2.4 mm) compared with patients in the low-grade group (involved knee: 2.0 ± 1.4 mm; side-to-side difference: 1.4 ± 1.5 mm) (both P <.01). As measured with the inertial sensor, tibial acceleration for patients in the high-grade group was significantly higher (involved knee: 7.2 ± 5.3 m/s2; side-to-side difference: 4.2 ± 5.4 m/s2) compared with patients in the low-grade group (involved knee: 4.2 ± 1.6 m/s2; side-to-side difference: 1.2 ± 1.2 m/s2) (both P <.01). Conclusion: The inertial sensor and image analysis techniques were able to detect differences between low- and high-grade pivot-shift test results. A quantitative assessment of the pivot-shift test could augment the diagnosis of an ACL injury and improve the ability to detect changes in rotatory knee laxity over time