Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding

We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made

Spectral Lags Obtained by CCF of Smoothed Lightcurves

We present a new technique to calculate the spectral lags of gamma-ray bursts (GRBs). Unlike previous processing methods, we first smooth the light curves of gamma-ray bursts in high and low energy bands using the "Loess" filter, then, we directly define the spectral lags as such to maximize the cross-correlation function (CCF) between two smoothed light curves. This method is suitable for various shapes of CCF; it effectively avoids the errors caused by manual selections for the fitting function and fitting interval. Using the method, we have carefully measured the spectral lags of individual pulses contained in BAT/Swift gamma-ray bursts with known redshifts, and confirmed the anti-correlation between the spectral lag and the isotropy luminosity. The distribution of spectral lags can be well fitted by four Gaussian components, with the centroids at 0.03 s, 0.09 s, 0.15 s, and 0.21 s, respectively. We find that some spectral lags of the multi-peak GRBs seem to evolve with time

Isometric embedding via strongly symmetric positive systems

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.Comment: 39 page

The accretion rate independence of horizontal branch oscillation in XTE J1701-462

We study the temporal and energy spectral properties of the unique neutron star low-mass X-ray binary XTE J1701-462. After assuming the HB/NB vertex as a reference position of accretion rate, the horizontal branch oscillation (HBO) of the HB/NB vertex is roughly 50 Hz. It indicates that the HBO is independent with the accretion rate or the source intensity. The spectral analysis shows $R_{\rm{in}}\propto\dot{M}_{\rm{Disk}}^{2.9\pm0.09}$ in the HB/NB vertex and $R_{\rm{in}}\propto\dot{M}_{\rm{Disk}}^{1.7\pm0.06}$ in the NB/FB vertex, which implies that different accretion rate may be produced in the HB/NB vertex and the NB/FB vertex. The Comptonization component could be fitted by constrained broken power law (CBPL) or nthComp. Different with GX 17+2, the frequencies of HBO positively correlate with the inner disk radius, which contradict with the prediction of Lense-Thirring precession model. XTE J1701-462, both in the Cyg-like phase and in the Sco-like phase, follows a positive correlation between the break frequency of broad band noise and the characteristic frequency of HBO, which is called the W-K relation. An anticorrelation between the frequency of HBO and photon energy is observed. Moreover, the rms of HBO increases with photon energy till ~10 keV. We discuss the possible origin of HBO from corona in XTE J1701-462.Comment: 45 pages, 18 figures, accepted by Ap