Phase-Amplitude Relations for a Particle with a Superposition of Two Energy Levels in a Double Potential Well.

We study the connection between the phase and the amplitude of the wave function and the conditions under which this relationship exists. For this we use the model of particle in a box. We have shown that the amplitude can be calculated from the phase and vice versa if the log analytical uncertainty relations are satisfied.

Heart Rate Variability for Risk Assessment of Myocardial Ischemia in Patients Without Known Coronary Artery Disease: The HRV-DETECT (Heart Rate Variability for the Detection of Myocardial Ischemia) Study.

Background Detecting significant coronary artery disease (CAD) in the general population is complex and relies on combined assessment of traditional CAD risk factors and noninvasive testing. We hypothesized that a CAD-specific heart rate variability (HRV) algorithm can be used to improve detection of subclinical or early ischemia in patients without known CAD. Methods and Results Between 2014 and 2018 we prospectively enrolled 1043 patients with low to intermediate pretest probability for CAD who were screened for myocardial ischemia in tertiary medical centers in the United States and Israel. Patients underwent 1-hour Holter testing, with immediate HRV analysis using the HeartTrends DyDx algorithm, followed by exercise stress echocardiography (n=612) or exercise myocardial perfusion imaging (n=431). The threshold for low HRV was identified using receiver operating characteristic analysis based on sensitivity and specificity. The primary end point was the presence of myocardial ischemia detected by exercise stress echocardiography or exercise myocardial perfusion imaging. The mean age of patients was 61 years and 38% were women. Myocardial ischemia was detected in 66 (6.3%) patients. After adjustment for CAD risk factors and exercise stress testing results, low HRV was independently associated with a significant 2-fold increased likelihood for myocardial ischemia (odds ratio, 2.00; 95% CI, 1.41-2.89 [P=0.01]). Adding HRV to traditional CAD risk factors significantly improved the pretest probability for myocardial ischemia. Conclusions Our data from a large prospective international clinical study show that short-term HRV testing can be used as a novel digital-health modality for enhanced risk assessment in low- to intermediate-risk individuals without known CAD. Clinical Trial Registration URL: http://www.ClinicalTrials.gov. Unique identifiers: NCT01657006, NCT02201017).

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.

Comparison of the usefulness of heart rate variability versus exercise stress testing for the detection of myocardial ischemia in patients without known coronary artery disease.

Heart rate variability (HRV) has been shown to be attenuated in patients with coronary artery disease (CAD) and may, therefore, be possibly used for the early detection of myocardial ischemia. We aimed to evaluate the diagnostic yield of a novel short-term HRV algorithm for the detection of myocardial ischemia in subjects without known CAD. We prospectively enrolled 450 subjects without known CAD who were referred to tertiary medical centers for exercise stress testing (EST) with single-photon emission computed tomography myocardial perfusion imaging (MPI). All subjects underwent 1-hour Holter testing with subsequent HRV analysis before EST with MPI. The diagnostic yield of HRV analysis was compared with EST, using MPI as the gold standard for the noninvasive detection of myocardial ischemia. All subjects had intermediate pretest probability for CAD. Mean age was 62 years, 38% were women, 51% had hypertension, and 25% diabetes mellitus. HRV analysis showed superior sensitivity (77%) compared with standard EST (27%). After multivariate adjustment, HRV was independently associated with an 8.4-fold (p <0.001) increased likelihood for the detection of myocardial ischemia by MPI, whereas EST did not show a statistically significant association with a positive MPI (odds ratio 2.1; p = 0.12). Of subjects who were referred for subsequent coronary angiography, the respective sensitivities of HRV and EST for the detection of significant CAD were 73% versus 26%. Our data suggest that HRV can be used as an important noninvasive technique for the detection of myocardial ischemia in subjects without known CAD, providing superior sensitivity to conventional EST in this population.

Heart rate variability density analysis (Dyx) for identification of appropriate implantable cardioverter defibrillator recipients among elderly patients with acute myocardial infarction and left ventricular systolic dysfunction.

AIMS: Dyx is a new heart rate variability (HRV) density analysis specifically designed to identify patients at high risk for malignant ventricular arrhythmias. The aim of this study was to test if Dyx can improve risk stratification for malignant ventricular tachyarrhythmias and to test if the previously identified cut-off can be reproduced. METHODS AND RESULTS: This study included 248 patients from the CARISMA study with ejection fraction ≤40% after an acute myocardial infarction and an analysable 24 h Holter recording. All patients received an implantable cardiac monitor, which was used to diagnose the primary endpoint of near-fatal or fatal ventricular tachyarrhythmias likely preventable by an implantable cardioverter defibrillator (ICD), during a period of 2 years. A Dyx ≤ 1.96 was considered abnormal. The secondary endpoint was cardiovascular death. At enrolment 59 patients (24%) had a Dyx ≤ 1.96 and 20 experienced a primary endpoint. A Dyx ≤ 1.96 was associated with a significantly increased risk for malignant arrhythmias [hazards ratio (HR) = 4.36 (1.81-10.52), P = 0.001] and cardiovascular death [HR = 3.47 (1.38-8.74), P = 0.008]. Compared with important clinical risk parameters (age >70 years and QRS > 120 ms), Dyx ≤ 1.96 significantly added predictive value (P = 0.0066). CONCLUSIONS: Dyx was a better predictor of ventricular tachyarrhythmias than the traditional measures of HRV and heart rate turbulence, particularly in the elderly. Dyx might be a useful tool for better selection of ICD candidates in the elderly population, since a normal Dyx in this group was associated with a very low risk for malignant ventricular arrhythmias.ClinicalTrials.gov Identifier NCT00145119.

Multipole analysis of heart rate variability as a predictor of imminent ventricular arrhythmias in ICD patients.

BACKGROUND: Contemporary implantable cardiac defibrillators (ICD) enable storage of multiple, preepisode R-R recordings in patients who suffered from ventricular tachyarrhythmia (VTA). Timely prediction of VTA, using heart rate variability (HRV) analysis techniques, may facilitate the implementation of preventive and therapeutic strategies. AIM: To evaluate the novel multipole method of the HRV analysis in prediction of imminent VTAs in ICD patients. METHODS: We screened patients from the Biotronik HAWAI Registry (Heart Rate Analysis with Automated ICDs). A total of 28 patients from the HAWAI registries (phase I and II), having medical records, who had experienced documented, verified VTA during the 2-year follow-up, were included in our analysis. HRV during preepisode recordings of 4,500 R-R intervals were analyzed using the Dyx parameter and compared to HRV of similar length recordings from the same patients that were not followed by arrhythmia. RESULTS: Our study population consisted mainly of men 25 of 28 (89%), average age of 64.8 ± 9.4 years, 92% with coronary artery disease. HRV during 64 preevent recordings (2.3 events per patient on average) was analyzed and compared with 60 control recordings. The multipole method of HRV analysis showed 50% sensitivity and 91.6% specificity for prediction of ventricular tachycardia/ventricular fibrillation in the study population, with 84.5% positive predictive value. No statistically significant correlation was found between various clinical parameters and the sensitivity of imminent VTA predetection in our patients. CONCLUSION: The multipole method of HRV analysis emerges as a highly specific, possible predictor of imminent VTA, providing an early warning allowing to prepare for an arrhythmic episode.

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.