We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.

We prove that for spacetimes solving the EinsteinMaxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In Nonlinear nature of gravitation and gravitational-wave experiments, Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational-wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 2.6 that for the EM equations this permanent displacement exhibits a term coming from the electromagnetic field. This term is at the same highest order as the purely gravitational term that governs the EV situation. On the other hand, in Section 3, we show that to leading order, the presence of the electromagnetic field

Gravitational waves are predicted by the general theory of relativity. It has been shown that gravitational waves have a nonlinear memory, displacing test masses permanently. This is called the Christodoulou memory. We proved that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves, enlarging the permanent displacement of test masses. In experiments like LISA or LIGO which measure distances of test masses, the Christodoulou memory will manifest itself as a permanent displacement of these objects. It has been suggested to detect the Christodoulou memory effect using radio telescopes investigating small changes in pulsar's pulse arrival times. The latter experiments are based on present-day technology and measure changes in frequency. In the present paper, we study the electromagnetic Christodoulou memory effect and compute it for binary neutron

We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in the affirmative.

In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on algebraic functions. By solving the Yamabe equation with the Ricci flow method, we can conformally map a brain surface to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of several types of anatomical surfaces in MRI scans of the brain, including the hippocampi and the cerebral cortices with various landmark curves labeled. For the cerebral cortical surfaces, we show the parameterization results are consistent with selected landmark curves and can be matched to each other using constrained harmonic maps. Unlike previous