Nonlinear softening as a predictive precursor to climate tipping

Approaching a dangerous bifurcation, from which a dynamical system such as the Earth's climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We consider an underlying potential, as defined theoretically for a saddle-node fold and (via averaging) for a Hopf bifurcation. Close to a stable state, this potential has a parabolic form; but approaching a jump it becomes increasingly dominated by softening nonlinearities. If we have already detected a decrease in the linear decay rate, nonlinear information allows us to estimate the propensity for early tipping due to noise. We argue that one needs to extract information about the nonlinear features (a "softening") of the underlying potential from the time series to judge the probability and timing of tipping. This analysis is the logical next step if one has detected a decrease of the linear decay rate. If there is no discernable trend in the linear analysis, nonlinear softening is even more important in showing the proximity to tipping. After extensive normal form calibration studies, we check two geological time series from paleo-climate tipping events for softening of the underlying well. For the ending of the last ice age, where we find no convincing linear precursor, we identify a statistically significant nonlinear softening towards increasing temperature. The analysis has thus successfully detected a warning of the imminent tipping event.Comment: 22 pages, 11 figures, changed title back, corrected smaller mistakes, updated reference

Unveiling fast field oscillations through comodulation

Phase-amplitude coupling analysis shows that theta phase modulates oscillatory activity not only within the traditional gamma band (30–100 Hz) but also at faster frequencies, called high-frequency oscillations (HFOs; 120–160 Hz). To date, however, theta-associated HFOs have been reported by only a small number of laboratories. Here we characterized coupling patterns during active waking (aWk) and rapid eye movement (REM) sleep in local field potentials (LFPs) from the parietal cortex and hippocampus of rats, focusing on how theta-associated HFOs can be detected. We found that electrode geometry and impedance only mildly influence HFO detection, whereas recording location and behavioral state are main factors. HFOs were most prominent in parietal cortex and during REM sleep, although they could also be detected in stratum oriens-alveus and during aWK. The underreporting of HFOs may thus be a result of higher prevalence of recordings from the pyramidal cell layer. However, at this layer, spike-leaked HFOs (SLHFOs) dominate, which represent spike contamination of the LFP and not genuine oscillations. In contrast to HFOs, high-gamma (HG; 60–100 Hz) coupled to theta below the pyramidal cell layer; theta–HG coupling increased during REM sleep. Theta also weakly modulated low-gamma (LG; 30–60 Hz) amplitude, mainly in the parietal cortex; theta–LG coupling did not vary between aWK and REM sleep. HG and HFOs were maximal near the theta peak, parietal LG at the ascending phase, hippocampal LG at the descending phase, and SLHFOs at the trough. Our results unveil four types of fast LFP activity coupled to theta and outline how to detect theta-associated HFOs

On admissibility criteria for weak solutions of the Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof

Universal Guard Problems

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities

Experiments in the marking of seals and sea-lions

This report reviews experiments in the marking, for study purposes, of seals, sea-lions, and fur seals in the North Atlantic, North Pacific, and Antarctic regions. Also discussed are the results of studies of the northern fur seal, especially the series from 1940 to 1049 carried out by U.S. Government agents on the Pribilof Islands, Alaska. (PDF contains 38 pages

North American Fauna 64

Examination of the body covering of the northern fur seal at each stage of growth as well as variations and anomalies