Entropy Measures as Geometrical Tools in the Study of Cosmology.

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson-Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence.

Heart Rate Variability Density Analysis (Dyx) and Prediction of Long-Term Mortality after Acute Myocardial Infarction.

AIMS: The density HRV parameter Dyx is a new heart rate variability (HRV) measure based on multipole analysis of the Poincaré plot obtained from RR interval time series, deriving information from both the time and frequency domain. Preliminary results have suggested that the parameter may provide new predictive information on mortality in survivors of acute myocardial infarction (MI). This study compares the prognostic significance of Dyx to that of traditional linear and nonlinear measures of HRV. METHODS AND RESULTS: In the Nordic ICD pilot study, patients with an acute MI were screened with 2D echocardiography and 24-hour Holter recordings. The study was designed to assess the power of several HRV measures to predict mortality. Dyx was tested in a subset of 206 consecutive Danish patients with analysable Holter recordings. After a median follow-up of 8.5 years 70 patients had died. Of all traditional and multipole HRV parameters, reduced Dyx was the most powerful predictor of all-cause mortality (HR 2.4; CI 1.5 to 3.8; P < 0.001). After adjustment for known risk markers, such as age, diabetes, ejection fraction, previous MI and hypertension, Dyx remained an independent predictor of mortality (P = 0.02). Reduced Dyx also predicted cardiovascular death (P < 0.01) and sudden cardiovascular death (P = 0.05). In Kaplan-Meier analysis, Dyx significantly predicted mortality in patients both with and without impaired left ventricular systolic function (P < 0.0001). CONCLUSION: The new nonlinear HRV measure Dyx is a promising independent predictor of mortality in a long-term follow-up study of patients surviving a MI, irrespectively of left ventricular systolic function.

Geometry of Hamiltonian chaos.

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.