The Rigidity and Gap Theorem for Liouville's Equation

In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying domain is a disc. More generally, we prove some gap theorems relating such a boundary integral to the number of components of the boundary. The conformal structure plays an essential role

Standard Model Higgs Searches at the Tevatron

We present the results of direct searches for the standard model Higgs boson at the Tevatron. Results are derived from the complete Tevatron Run II dataset, with a measured integrated luminosity of 10 fb$^{-1}$ of proton-antiproton data. The searches are performed for assumed Higgs masses between 90 and 200 GeV/c$^2$. We observe an excess of events in the data compared with the background predictions, which is most significant in the mass range between 115 and 135 GeV/c$^2$, consistent with the Higgs-like particle recently observed by ATLAS and CMS. The largest local significance is 2.7 standard deviations, corresponding to a global significance of 2.2 standard deviations. We also combine separate searches for $H\rightarrow b\bar b$ and $H\rightarrow W^+W^-$, and find that the excess is concentrated in the $H\rightarrow b\bar b$ channel, although the results in the $H\rightarrow W^+W^-$ channel are still consistent with the possible presence of a low-mass Higgs boson.Comment: This is a write up based on the talk I gave at the phenomenology 2012 symposium, 7-9 May 2012, University of Pittsburgh. 10 pages, 14 figure

Tevatron Combination and Higgs Boson Properties

We present the Tevatron combination of searches for the Higgs boson and studies of its properties. The searches use up to 10 fb$^{-1}$ of Tevatron collider Run II data. We observe a significant excess of events in the mass range between 115 and 140 GeV/c$^2$. The local significance corresponds to 3 Gaussian standard deviations at the mass of 125 GeV/c$^2$. Furthermore, we separately combine searches for the Higgs boson decaying to $b\bar b$, $\tau^+\tau^-$, $W^+W^-$, and photon pairs in the final states. The observed signal strengths in all channels are consistent with the presence of a standard model scalar boson with a mass of 125 GeV/c$^2$. Studies of the couplings at the Tevatron are consistent with SM predictions and are complementary to those performed at LHC.Comment: Submitted to the Proceeding of the XLVIIIth Recontres de Moriond electroweak session, 8 pages, 21 figure

Steganographic Codes -- a New Problem of Coding Theory

To study how to design steganographic algorithm more efficiently, a new coding problem -- steganographic codes (abbreviated stego-codes) -- is presented in this paper. The stego-codes are defined over the field with $q(q\ge2)$ elements. Firstly a method of constructing linear stego-codes is proposed by using the direct sum of vector subspaces. And then the problem of linear stego-codes is converted to an algebraic problem by introducing the concept of $t$th dimension of vector space. And some bounds on the length of stego-codes are obtained, from which the maximum length embeddable (MLE) code is brought up. It is shown that there is a corresponding relation between MLE codes and perfect error-correcting codes. Furthermore the classification of all MLE codes and a lower bound on the number of binary MLE codes are obtained based on the corresponding results on perfect codes. Finally hiding redundancy is defined to value the performance of stego-codes.Comment: 7 pages with 1 figur

Infinitely many solutions for a nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}),$$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.Comment: 39fages, 0 figures. arXiv admin note: text overlap with arXiv:1210.8209, arXiv:1209.2824 by other author

"Ge Shu Zhi Zhi": Towards Deep Understanding about Worlds

"Ge She Zhi Zhi" is a novel saying in Chinese, stated as "To investigate things from the underlying principle(s) and to acquire knowledge in the form of mathematical representations". The saying is adopted and modified based on the ideas from the Eastern and Western philosophers. This position paper discusses the saying in the background of artificial intelligence (AI). Some related subjects, such as the ultimate goals of AI and two levels of knowledge representations, are discussed from the perspective of machine learning. A case study on objective evaluations over multi attributes, a typical problem in the filed of social computing, is given to support the saying for wide applications. A methodology of meta rules is proposed for examining the objectiveness of the evaluations. The possible problems of the saying are also presented.Comment: 10 pages, in Chinese. 5 figures, 2 table

On the Unicity Distance of Stego Key

Steganography is about how to send secret message covertly. And the purpose of steganalysis is to not only detect the existence of the hidden message but also extract it. So far there have been many reliable detecting methods on various steganographic algorithms, while there are few approaches that can extract the hidden information. In this paper, the difficulty of extracting hidden information, which is essentially a kind of privacy, is analyzed with information-theoretic method in the terms of unicity distance of steganographic key (abbreviated stego key). A lower bound for the unicity distance is obtained, which shows the relations between key rate, message rate, hiding capacity and difficulty of extraction. Furthermore the extracting attack to steganography is viewed as a special kind of cryptanalysis, and an effective method on recovering the stego key of popular LSB replacing steganography in spatial images is presented by combining the detecting technique of steganalysis and correlation attack of cryptanalysis together. The analysis for this method and experimental results on steganographic software ``Hide and Seek 4.1" are both accordant with the information-theoretic conclusion.Comment: 8 pages, 3 figure

The Loewner-Nirenberg problem in singular domains

We study the asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates

On spectral properties of high-dimensional spatial-sign covariance matrices in elliptical distributions with applications

Spatial-sign covariance matrix (SSCM) is an important substitute of sample covariance matrix (SCM) in robust statistics. This paper investigates the SSCM on its asymptotic spectral behaviors under high-dimensional elliptical populations, where both the dimension $p$ of observations and the sample size $n$ tend to infinity with their ratio $p/n\to c\in (0, \infty)$. The empirical spectral distribution of this nonparametric scatter matrix is shown to converge in distribution to a generalized Mar\v{c}enko-Pastur law. Beyond this, a new central limit theorem (CLT) for general linear spectral statistics of the SSCM is also established. For polynomial spectral statistics, explicit formulae of the limiting mean and covarance functions in the CLT are provided. The derived results are then applied to an estimation procedure and a test procedure for the spectrum of the shape component of population covariance matrices

Reasoning about Cardinal Directions between Extended Objects: The Hardness Result

The cardinal direction calculus (CDC) proposed by Goyal and Egenhofer is a very expressive qualitative calculus for directional information of extended objects. Early work has shown that consistency checking of complete networks of basic CDC constraints is tractable while reasoning with the CDC in general is NP-hard. This paper shows, however, if allowing some constraints unspecified, then consistency checking of possibly incomplete networks of basic CDC constraints is already intractable. This draws a sharp boundary between the tractable and intractable subclasses of the CDC. The result is achieved by a reduction from the well-known 3-SAT problem.Comment: 24 pages, 24 figure