12,794 research outputs found
Dependence Structure Analysis Of Meta-level Metrics in YouTube Videos: A Vine Copula Approach
This paper uses vine copula to analyze the multivariate statistical
dependence in a massive YouTube dataset consisting of 6 million videos over 25
thousand channels. Specifically we study the statistical dependency of 7
YouTube meta-level metrics: view count, number of likes, number of comments,
length of video title, number of subscribers, click rates, and average
percentage watching. Dependency parameters such as the Kendall's tau and tail
dependence coefficients are computed to evaluate the pair-wise dependence of
these meta-level metrics. The vine copula model yields several interesting
dependency structures. We show that view count and number of likes' are in the
central position of the dependence structure. Conditioned on these two metrics,
the other five meta-level metrics are virtually independent of each other.
Also, Sports, Gaming, Fashion, Comedy videos have similar dependence structure
to each other, while the News category exhibits a strong tail dependence. We
also study Granger causality effects and upload dynamics and their impact on
view count. Our findings provide a useful understanding of user engagement in
YouTube
General Matrix-Valued Inhomogeneous Linear Stochastic Differential Equations and Applications
The expressions of solutions for general matrix-valued
inhomogeneous linear stochastic differential equations are derived. This
generalizes a result of Jaschke (2003) for scalar inhomogeneous linear
stochastic differential equations. As an application, some vector-valued
inhomogeneous nonlinear stochastic differential equations are reduced to random
differential equations, facilitating pathwise study of the solutions
Approximation of the inertial manifold for a nonlocal dynamical system
We consider inertial manifolds and their approximation for a class of partial
differential equations with a nonlocal Laplacian operator
, with . The nonlocal or fractional
Laplacian operator represents an anomalous diffusion effect. We first establish
the existence of an inertial manifold and highlight the influence of the
parameter . Then we approximate the inertial manifold when a small
normal diffusion (with ) enters
the system, and obtain the estimate for the Hausdorff semi-distance between the
inertial manifolds with and without normal diffusion.Comment: 19page
Random data Cauchy problem for a generalized KdV equation in the supercritical case
We consider the Cauchy problem for a generalized KdV equation
\begin{eqnarray*}
u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0,
\end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl.
Math.46(1993), 527-620)proved that the problem is globally well-posed in
H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical
regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc.
53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the
time of existence T and the continuous dependence cannot be expressed in terms
of the size of the data in the H^{\frac{3}{14}}-norm. In this present paper, we
prove that almost sure local in time well-posedness holds in H^{s}(\R) with
s>\frac{17}{112}, whose lower bound is below \frac{3}{14}.
The key ingredients are the Wiener randomization of the initial data and
probabilistic Strichartz estimates together with some important embedding
Theorems.Comment: 44page
Generalized Radial Equations in a Quantum N-Body Problem
We demonstrate how to separate the rotational degrees of freedom in a quantum
N-body problem completely from the internal ones. It is shown that any common
eigenfunction of the total orbital angular momentum () and the parity in
the system can be expanded with respect to base-functions, where
the coefficients are the functions of the internal variables. We establish
explicitly the equations for those functions, called the generalized radial
equations, which are coupled partial differential equations
containing only internal variables.Comment: 7 pages, no figure, RevTe
Quiver mutations and Boolean reflection monoids
In 2010, Everitt and Fountain introduced the concept of reflection monoids.
The Boolean reflection monoids form a family of reflection monoids (symmetric
inverse semigroups are Boolean reflection monoids of type ). In this paper,
we give a family of presentations of Boolean reflection monoids and show how
these presentations are compatible with quiver mutations of orientations of
Dynkin diagrams with frozen vertices. Our results recover the presentations of
Boolean reflection monoids given by Everitt and Fountain and the presentations
of symmetric inverse semigroups given by Popova respectively. Surprisingly,
inner by diagram automorphisms of irreducible Weyl groups and Boolean
reflection monoids can be constructed by sequences of mutations preserving the
same underlying diagrams. Besides, we show that semigroup algebras of Boolean
reflection monoids are cellular algebras.Comment: 33 pages, final version, to appear in Journal of Algebr
Cluster algebras and snake modules
Snake modules introduced by Mukhin and Young form a family of modules of
quantum affine algebras. The aim of this paper is to prove that the
Hernandez-Leclerc conjecture about monoidal categorifications of cluster
algebras is true for prime snake modules of types and . We prove
that prime snake modules are real. We introduce -systems consisting of
equations satisfied by the -characters of prime snake modules of types
and . Moreover, we show that every equation in the -system of
type (respectively, ) corresponds to a mutation in the cluster
algebra (respectively, ) constructed by Hernandez
and Leclerc and every prime snake module of type (respectively, )
corresponds to some cluster variable in (respectively,
). In particular, this proves that the Hernandez-Leclerc
conjecture is true for all prime snake modules of types and
Random data Cauchy problem for the wave equation on compact manifold
Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008),
449-475.), firstly, we construct the local strong solution to the cubic
nonlinear wave equation with random data for a large set of initial data in
with , where M is a three dimensional compact
manifold with boundary, moreover, our result improves the result of Theorem 2
in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strong
solution to the quintic nonlinear wave equation with random data for a large
set of initial data in with , where M is a two
dimensional compact boundaryless manifold; finally, we construct the local
strong solution to the quintic nonlinear wave equation with random data for a
large set of initial data in with , where M is
a two dimensional compact manifold with boundary.Comment: We correct some misprint
Probabilistic pointwise convergence problem of some dispersive equations
In this paper, we investigate the almost surely pointwise convergence problem
of free KdV equation, free wave equation, free elliptic and non-elliptic
Schr\"odinger equation respectively. We firstly establish some estimates
related to the Wiener decomposition of frequency spaces which are just Lemmas
2.1-2.6 in this paper. Secondly, by using Lemmas 2.1-2.6, 3.1, we establish the
probabilistic estimates of some random series which are just Lemmas 3.2-3.11 in
this paper. Finally, combining the density theorem in L with Lemmas
3.2-3.11, we obtain almost surely pointwise convergence of the solutions to
corresponding equations with randomized initial data in , which require
much less regularity of the initial data than the rough data case. At the same
time, we present the probabilistic density theorem, which is Lemma 3.11 in this
paper
The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces
The goal of this paper is three-fold. Firstly, we prove that the Cauchy
problem for generalized KP-I equation \begin{eqnarray*}
u_{t}+|D_{x}|^{\alpha}\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4
\end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces with and .
Secondly, we prove that the problem is globally well-posed in
with
if .
Finally, we prove that the problem is globally well-posed in
with if
. Our result improves the result of Saut and Tzvetkov (J. Math. Pures
Appl. 79(2000), 307-338.) and Li and Xiao (J. Math. Pures Appl. 90(2008),
338-352.).Comment: 57 page
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