Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form V(x) = ||x||pQ := (xt>Qx) p/2, where the parameters are the positive definite matrix Q and the number p > 0. We give several examples of our proposed method and show how it improves previous results

Simplicial complex with approximate rotational symmetry: a general class of simplicial complexes

We study the transformation of the vertices of a certain simple simplicial complex in n-dimensional Euclidian space and prove that the resulting set of simplices is a simplicial complex with an approximate rotational symmetry. Such simplicial complexes have applications in computing Lyapunov function for nonlinear dynamical systems using linear optimization and are also of interest for other applications

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

Movement does not promote recovery of motor output following acute experimental muscle pain

Objective. To examine the effect of motor activity on the magnitude and duration of altered corticomotor output following experimental muscle pain. Design. Experimental, pre-post test. Setting. University laboratory. Subjects. Twenty healthy individuals. Methods. Participants were randomly allocated to a Rest or Movement group. The Rest group sat quietly without moving for the duration of the experiment. The Movement group repeated a unimanual pattern of five sequential keystrokes as quickly and as accurately as possible immediately following the resolution of pain. Pain was induced into the right extensor carpi radialis brevis muscle by a bolus injection of 0.5 mL hypertonic saline. Corticomotor output was assessed as motor evoked potentials in response to transcranial magnetic stimulation before, immediately after, and at 10, 20, and 30 minutes following pain resolution. Pain intensity was recorded every 30 seconds using an 11-point numerical rating scale. Results. There was no difference in peak pain intensity (P < 0.09) or duration (P < 0.2) between groups. Corticomotor output was reduced in both groups (P < 0.002) at 10 minutes (P < 0.002), 20 minutes (P < 0.02), and 30 minutes (P < 0.037) following the resolution of pain relative to baseline. There was no difference between groups at any time point. Conclusions. Performance of motor activity immediately following the resolution of acute muscle pain did not alter the magnitude or duration of corticomotor depression. Understanding corticomotor depression in the postpain period and what factors promote recovery has relevance for clinical pain syndromes where ongoing motor dysfunction, in the absence of pain, may predispose to symptom persistence or recurrence

Six commonly used empirical body surface area formulas disagreed in young children undergoing corrective heart surgery.

To access publisher's full text version of this article, please click on the hyperlink in Additional Links field or click on the hyperlink at the top of the page marked DownloadAim: Formulas for empirical body surface area (BSA), which is used to estimate body size and standardise physiological parameters, may disagree in children. We compared six commonly used BSA formulas-Du Bois, Boyd, Costeff, Haycock, Meban and Mosteller-in a surgical cohort. Methods: This retrospective single-centre cohort study comprised 68 children who had corrective heart surgery at Skåne University Children's Hospital, Lund, Sweden, from February 2010 to March 2017. Results: The children (51% female) underwent surgery at a mean weight of 7.0 kilograms (range 2.7-14.1 kg) and a mean age 11 months (range 0-43 months). All the BSA formulas showed good correlation with mean BSA, but there were considerable variations between them. Mosteller's formula was exactly the same as the mean BSA (bias 0.000). The Du Bois and Boyd formulas had the largest mean BSA deviations (bias -0.012 and 0.015). Costeff's formula showed good agreement with mean BSA, Haycock's formula showed minimal overestimation and Meban's formula demonstrated a systemic error in older children. Conclusion: Commonly used BSA formulas did not agree in young children undergoing heart surgery, but they were all close to the overall mean of the six formulas, with the Mosteller formula producing the same value