No functions continuous only at points in a countable dense set

We give a short proof that if a function is continuous on a countable dense set, then it is continuous on an uncountable set. This is done for functions defined on nonempty complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra's theorem and the history of this problem. We give a simple example, for each complete metric space without isolated points and each countable subset, of a real-valued function that is discontinuous only on that subset.Comment: Expanded the result and added historical references and discussio

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space \H^n ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{array}{rcl} \ds \int_\Sigma \s_4 d \mu\ge \ds\vs C_{n-1}^4\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 12 + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array} provided that $\Sigma$ is a horospherical convex hypersurface. Equality holds if and only if $\Sigma$ is a geodesic sphere in \H^n.Comment: 18page

A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $\tau$ satisfies $\tau > \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.Comment: 32 pages. arXiv:1211.7305 was integrated into this new version as an applicatio

The Gauss-Bonnet-Chern mass of conformally flat manifolds

In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.Comment: 17 pages, references added, the statement of Prop. 4.6 correcte

Bimodal waveguide interferometer RI sensor fabricated on low-cost polymer platform

A refractive index sensor based on bimodal waveguide interferometer is demonstrated on the low-cost polymer platform for the first time. Different from conventional interferometers which make use of the interference between the light from two arms, bimodal waveguide interferometers utilize the interference between the two different internal modes in the waveguide. Since the utilized first higher mode has a wide evanescent tail which interacts with the external environment, the interferometer can reach a high sensitivity. Instead of vertical bimodal structure which is normally employed, the lateral bimodal waveguide is adopted in order to simplify the fabrication process. A unique offset between the centers of single mode waveguide and bimodal waveguide is designed to excite the two different modes with equal power which contributes to the maximum fringe visibility. The bimodal waveguide interferometer is finally fabricated on optical polymer (Ormocore) which is transparent at both infrared and visible wavelengths. It is fabricated using the UV-based soft imprint technique which is simple and reproductive. The bulk sensitivity of fabricated interferometer sensor with a 5 mm sensing length is characterized using different mass concentration sodium chloride solutions. The sensitivity is obtained as 316 pi rad/RIU and the extinction ratio can reach 18 dB