A unified approach is presented for establishing a broad class of Cram\'er-Rao inequalities for the location parameter, including, as special cases, the original inequality of Cram\'er and Rao, as well as an $L^p$ version recently established by the authors. The new approach allows for generalized moments and Fisher information measures to be defined by convex functions that are not necessarily homogeneous. In particular, it is shown that associated with any log-concave random variable whose density satisfies certain boundary conditions is a Cram\'er-Rao inequality for which the given log-concave random variable is the extremal. Applications to specific instances are also provided.

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the nonintegrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface f admits singularities but its Gauss map  is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.

We consider minimal surfaces M which are complete, embedded, and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation
u+ f(u) = 0 in R3. Here f = −W0 with W bi-stable and balanced, for instance W(u) = 1 4 (1 − u2)2. We assume that M has m  2 ends, and additionally that M is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman- Meeks surface of any genus). We prove that for any small  > 0, the Allen-Cahn equation has a family of bounded solutions depending on m − 1 parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface M := −1M, with ends possibly diverging logarithmically from M. We prove that these solutions are L1-non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

entangled states. We construct sets of fewer than d orthogonal maximally entangled states which are not distinguished by one-way local operations and classical communication (LOCC) in the Hilbert space of d ⊗ d. The proof, based on the Fourier transform of an additive group, is very simple but quite effective. Simultaneously, our results give a general unified upper bound for the minimum number of one-way LOCC indistinguishable maximally entangled states. This improves previous results which only showed sets of N  d − 2 such states. Finally, our results also show that previous conjectures in Zhang et al. [Z.-C. Zhang, Q.-Y. Wen, F. Gao, G.-J. Tian, and T.-Q. Cao, Quant. Info. Proc. 13, 795 (2014)] are indeed correct.