This paper proves Birch and Swinnerton-Dyer conjecture for Q-curves defined over real quadratic number field with everywhere good reductions.

No abstract uploaded!

Motivated by the nuclear magnetic resonance (NMR) spectroscopy of biofluids (urine and blood serum), we present a recursive blind source separation (rBSS) method for nonnegative and correlated data. BSS problem arises when one attempts to recover a set of source signals from a set of mixture signals without knowing the mixing process. Various approaches have been developed to solve BSS problems relying on the assumption of statistical independence of the source signals. However, signal independence is not guaranteed in many real-world data like the NMR spectra of chemical compounds. The rBSS method introduced in this paper deals with the nonnegative and correlated signals arising in NMR spectroscopy of biofluids. The statistical independence requirement is replaced by a constraint which requires dominant interval(s) from each source signal over some of the other source signals in a

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

Consider electromagnetic waves in two-dimensional honeycomb structured media. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L^A=-\nabla_x·A(x) \nabla_x, where A(x) is Λ_h− periodic (Λ_h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is PC− invariant (A(x)=\overline{A(-x)}) and 120° rotationally invariant (A(R*x)=R*A(x)R, where R is a 120° rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schrödinger operators.