Global Solution for the incompressible Navier-Stokes equations] { Global Solution for the incompressible Navier-Stokes equations with a class of large data in BMO−1(R3)BMO^{-1}(\mathbb{R}^3)

In this paper, we shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$

Formulation of finite-time singularity for free-surface Euler equations

We give an extremely short proof that the free-surface incompressible, irrotational Euler equations with regular initial condition can form a finite time singularity in 2D or 3D. Thus, we provide a simple view of the problem studied by Castro, Cordoba, Fefferman, Gancedo, Lopez-Fernadez, Gomez-Serrano and Coutand, Shkoller.Comment: 6 page

A Set Theoretic Approach for Knowledge Representation: the Representation Part

In this paper, we propose a set theoretic approach for knowledge representation. While the syntax of an application domain is captured by set theoretic constructs including individuals, concepts and operators, knowledge is formalized by equality assertions. We first present a primitive form that uses minimal assumed knowledge and constructs. Then, assuming naive set theory, we extend it by definitions, which are special kinds of knowledge. Interestingly, we show that the primitive form is expressive enough to define logic operators, not only propositional connectives but also quantifiers.Comment: This paper targets an ambitious goal to rebuild a foundation of knowledge representation based on set theory rather than classical logic. Any comments are welcom

Cauchy problem of nonlinear Schr\"odinger equation with Cauchy problem of nonlinear Schr\"odinger equation with initial data in Sobolev space Ws,pW^{s,p} for p<2p<2

In this paper, we consider in $R^n$ the Cauchy problem for nonlinear Schr\"odinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\frac{2n}{n+1}n(1-\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.Comment: 12 page

Structured Production System (extended abstract)

In this extended abstract, we propose Structured Production Systems (SPS), which extend traditional production systems with well-formed syntactic structures. Due to the richness of structures, structured production systems significantly enhance the expressive power as well as the flexibility of production systems, for instance, to handle uncertainty. We show that different rule application strategies can be reduced into the basic one by utilizing structures. Also, many fundamental approaches in computer science, including automata, grammar and logic, can be captured by structured production systems

An Implication of Ether Drift

The experimental results of the two-photon absorption(TPA) and M\"{o}ssbauer-rotor(MR) for testing the isotropy of the speed of light are explained in an ether drift model with a drift velocity of $\sim 10^{-3}c$. Further tests of the ether drift assumption are suggested.Comment: 6 pages,2 postscript figure

Extracting Top Quark CP Violating Dipole Couplings via ttˉγt\bar t\gamma and ttˉZt\bar tZ Productions at the LHC

We propose to extract the electric and weak dipole moments of the top quark via $t\bar t\gamma$ and $t\bar t Z$ productions at the CERN LHC. With the large numbers of events available at the LHC, these dipole moments can be measured to the accuracy of $10^{-18}e cm$.Comment: 7 pages, 1 postscript figur

Set-based differential covariance testing for high-throughput data

The problem of detecting changes in covariance for a single pair of features has been studied in some detail, but may be limited in importance or general applicability. In contrast, testing equality of covariance matrices of a {\it set} of features may offer increased power and interpretability. Such approaches have received increasing attention in recent years, especially in the context of high-dimensional testing. These approaches have been limited to the two-sample problem and involve varying assumptions on the number of features $p$ vs. the sample size $n$. In addition, there has been little discussion of the motivating principles underlying various choices of statistic, and no general approaches to test association of covariances with a continuous outcome. We propose a uniform framework to test association of covariance matrices with an experimental variable, whether discrete or continuous. We describe four different summary statistics, to ensure power and flexibility under various settings, including a new "connectivity" statistic that is sensitive to changes in overall covariance magnitude. The approach is not limited by the data dimensions, and is applicable to situations where $p >> n$. For several statistics we obtain asymptotic $p$-values under relatively mild conditions. For the two-sample special case, we show that the proposed statistics are permutationally equivalent or similar to existing proposed statistics. We demonstrate the power and utility of our approaches via simulation and analysis of real data.Comment: arXiv admin note: substantial text overlap with arXiv:1609.0073

Critical Points of Neural Networks: Analytical Forms and Landscape Properties

Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that the analytical forms of the critical points characterize the values of the corresponding loss functions as well as the necessary and sufficient conditions to achieve global minimum. Furthermore, we exploit the analytical forms of the critical points to characterize the landscape properties for the loss functions of these neural networks. One particular conclusion is that: The loss function of linear networks has no spurious local minimum, while the loss function of one-hidden-layer nonlinear networks with ReLU activation function does have local minimum that is not global minimum

Global Low Regularity Solutions of Quasi-linear Wave Equations

In this paper we prove the global existence and uniqueness of the low regularity solutions to the Cauchy problem of quasi-linear wave equations with radial symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method.Comment: We prove global existence and uniqueness of low regularity solutions for quasi-linear wave equations in radial symmetric case. 53 pages, 3 figure