39,188 research outputs found
Global Solution for the incompressible Navier-Stokes equations] { Global Solution for the incompressible Navier-Stokes equations with a class of large data in
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
. This improves the classical result of Koch \&
Tataru \cite{koch-tataru}, for the global well-posedness with small initial
data
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 for
In this paper, we consider in the Cauchy problem for nonlinear
Schr\"odinger equation with initial data in Sobolev space for .
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 for . Moreover,
we show that in one space dimension, the problem is locally well posed in
for any .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 .
Further tests of the ether drift assumption are suggested.Comment: 6 pages,2 postscript figure
Extracting Top Quark CP Violating Dipole Couplings via and Productions at the LHC
We propose to extract the electric and weak dipole moments of the top quark
via and 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 .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 vs. the sample size . 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 . For several statistics we obtain asymptotic -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
Blow up of Solutions to Semilinear Wave Equations with variable coefficients and boundary
This paper is devoted to studying the following two initial-boundary value
problems for semilinear wave equations with variable coefficients on exterior
domain with subcritical exponent in space dimensions:
u_{tt}-partial_{i}(a_{ij}(x)\partial_{j}u)=|u|^{p}, (x,t)\in
\Omega^{c}\times(0,+\infty), n\geq 3 and
u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in
\Omega^{c}\times (0,+\infty), n\geq 1, where p 1<p<p_{1}(n)p \leq p_{2}(n)p_{1}(n)$ is the larger root of the quadratic equation
(n-1)p^{2}-(n+1)p-2=0, and p_{2}(n)=\frac{2}{n-1}+1, respectively. It is
well-known that the numbers p_{1}(n) and p_{2}(n) are the critical exponents.
We will establish two blowup results for the above two initial-boundary value
problems, it is proved that there can be no global solutions no matter how
small the initial data are, and also we give the lifespan estimate of solutions
for above problems
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