On the Willmore functional of 2-tori in some product Riemannian manifolds

We discuss the minimum of Willmore functional of torus in a Riemannian manifold $N$, especially for the case that $N$ is a product manifold. We show that when $N=S^2\times S^1$, the minimum of $W(T^2)$ is 0, and when $N=R^2\times S^1$, there exists no torus having least Willmore functional. When $N=H^2(-c)\times S^1$, and $x=\gamma\times S^1$, the minimum of $W(x)$ is $2\pi^2\sqrt{c}$.Comment: 12 page

Construction of Willmore two-spheres via harmonic maps into SO+(1,n+3)/(SO+(1,1)×SO(n+2))SO^+(1,n+3)/(SO^+(1,1)\times SO(n+2))

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by H\'elein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valued meromorphic 1-forms) of totally isotropic Willmore two-spheres in terms of the isotropic harmonic maps. In particular, the corresponding isotropic harmonic maps are of finite uniton type. The proof also contains a concrete way to construct examples of totally isotropic Willmore two-spheres and their adjoint transforms. As illustrations, two kinds of examples are obtained this way.Comment: 24 page

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\R^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\R^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\R^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a "constant lightlike vector". Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in $\R^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.Comment: 20 pages. Revised Versio

On Willmore surfaces in S^n of flat normal bundle

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal bundle, which improve a result of K. Yang. Then we derived that every Willmore two sphere with flat normal bundle in $S^n$ is conformal to a minimal surface with embedded planer ends in $\mathbb{R}^3$. We also point out that for a class of Willmore tori, they have flat normal bundle if and only if they locate in some $S^3$. In the end, we show that a Willmore surface with flat normal bundle must locate in some $S^6$Comment: 14 pages, all comments are welcom

Uniqueness of positive solutions with Concentration for the Schr\"odinger-Newton problem

We are concerned with the following Schr\"odinger-Newton problem \begin{equation} -\varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{|x-\xi|}d\xi\big)u,~x\in \mathbb R^3. \end{equation} For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schr\"odinger-Newton problem is quite different from those of Schr\"odinger equations

Towards the Adoption of Anti-spoofing Protocols

Email spoofing is a critical step of phishing, where the attacker impersonates someone the victim knows or trusts. In this paper, we conduct a qualitative study to explore why email spoofing is still possible after years of efforts to develop and deploy anti-spoofing protocols (e.g., SPF, DKIM, DMARC). First, we measure the protocol adoption by scanning 1 million Internet domains. We find the adoption rates are still low, especially for the new DMARC (3.1%). Second, to understand the reasons behind the low-adoption rate, we collect 4293 discussion threads (25.7K messages) from the Internet Engineering Task Force (IETF), a working group formed to develop and promote Internet standards. Our analysis shows key security and usability limitations in the protocol design, which makes it difficult to generate a positive "net effect" for a wide adoption. We validate our results by interviewing email administrators and discuss key implications for future anti-spoofing solutions

Superluminal telecommunication: an observable contradiction between quantum entanglement and relativistic causality

I present a schema for a superluminal telecommunication system based on polarization entangled photon pairs. Binary signals can be transmitted at superluminal speed in this system, if entangled photon pairs can really be produced. The existence of the polarization entangled photon pairs is in direct contradiction to the relativistic causality in this telecommunication system. This contradiction implies the impossibility of generating entangled photon pairs.Comment: 6 pages, 3 figure

Varieties of Dirac equation and flavors of leptons and quarks

I show that there exist twelve independent Dirac equations for spin 1/2 fermions. The Dirac fields that satisfy these equations can be grouped into six pairs according to the way they transform under continuous space-time transformations. These six pairs of Dirac equations correspond to the three quark generations and the three lepton generations. The charged V-A currents can be formed only from fields of the same pair. This property of the Dirac fields implies that a quark or lepton may be transformed only into its partner of the same generation via the charged-current weak interaction. According to the properties of the charged-current weak interaction, I conclude that different elementary fermion fields must satisfy different Dirac equations, and there may not be more than twelve flavors of elementary fermions that are already known.Comment: 5 pages, RevTe

Spinor equation for the W±W^{\pm} boson

I introduce spinor equations for the $W^{\pm}$ fields. The properties of these spinor equations under space-time transformation and under charge conjugation are studied. The expressions for electric charge and current and densities of the $W^{\pm}$ fields are obtained. Covariant quantization conditions are established, and the vacuum energy for the $W^{\pm}$ fields is found to be zero

A note on logistic regression and logistic kernel machine models

This is a note on logistic regression models and logistic kernel machine models. It contains derivations to some of the expressions in a paper -- SNP Set Analysis for Detecting Disease Association Using Exon Sequence Data -- submitted to BMC proceedings by these authors