arXiv subjects
Publications and source records attributed to Jie Wu.
We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. This result extends in some ways a classical result of Lions and Malgrange [12] and a recent result of the authors [10].
It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ in $\mathbb{R}^n_+\times[0,T]$ and $u(x,0)=0$ in $\mathbb{R}^n_+$ under certain conditions on $\{a^{ij}\}$ must vanish identically in $\mathbb{R}^n_+\times[0,T]$. The main point of the result is that the conditions imposed on $\{a^{ij}\}$ are of the type: $\{a^{ij}\}$ are Lipschitz and $|\nabla_xa^{ij}(x,t)|\leq \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.
The recent introduction of depth cameras like Leap Motion Controller allows researchers to exploit the depth information to recognize hand gesture more robustly. This paper proposes a novel hand gesture recognition system with Leap Motion Controller. A series of features are extracted from Leap Motion tracking data, we feed these features along with HOG feature extracted from sensor images into a multi-class SVM classifier to recognize performed gesture, dimension reduction and feature weighted fusion are also discussed. Our results show that our model is much more accurate than previous work.
Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for the number of rational points of bounded height on singular cubic hypersurfaces $S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in two aspects: first, it can be viewed as a modest start on the study of density of rational points on those singular cubic hypersurfaces which are not covered by the classical theorems of Davenport or Heath-Brown; second, it proves Manin's conjecture for singular cubic hypersurfaces $S_n$ defined above.
We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.
In this paper, we propose an alternative approach to implement the contact angle boundary condition on immersed surfaces for phase-field simulations of two-phase flows using the Cahn-Hilliard equation on a Cartesian mesh. This simple and effective method was inspired by previous works on the geometric formulation of the wetting boundary condition. In two dimensions, by making full use of the hyperbolic tangent profile of the order parameter, we were able to obtain its unknown value at a ghost point from the information at only one point in the fluid. This is in contrast with previous approaches using interpolations involving several points. The special feature allows this method to be easily implemented on immersed surfaces (including curved ones) that cut through the grid lines. It is verified through the study of two examples: (1) the shape of a drop on a circular cylinder with different contact angles; (2) the spreading of a drop on an embedded inclined wall with a given contact angle.
For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.
With the rapid development of software defined networking and network function virtualization, researchers have proposed a new cloud networking model called Network-as-a-Service (NaaS) which enables both in-network packet processing and application-specific network control. In this paper, we revisit the problem of achieving network energy efficiency in data centers and identify some new optimization challenges under the NaaS model. Particularly, we extend the energy-efficient routing optimization from single-resource to multi-resource settings. We characterize the problem through a detailed model and provide a formal problem definition. Due to the high complexity of direct solutions, we propose a greedy routing scheme to approximate the optimum, where flows are selected progressively to exhaust residual capacities of active nodes, and routing paths are assigned based on the distributions of both node residual capacities and flow demands. By leveraging the structural regularity of data center networks, we also provide a fast topology-aware heuristic method based on hierarchically solving a series of vector bin packing instances. Our simulations show that the proposed routing scheme can achieve significant gain on energy savings and the topology-aware heuristic can produce comparably good results while reducing the computation time to a large extent.
We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [14]. The cases of modular forms of half-integral weight and Siegel eigenforms are also considered.
In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.
A parameter, called the degree of diffraction, is defined to describe the diffractive spreading of a monochromatic light beam. The same as the degree of paraxiality that was introduced by Gawhary and Severini in Opt. Lett. 33, 1360 (2008), the degree of diffraction depends only on beam's angular spectrum. With this definition, it is possible to quantitatively compare the diffractive spreading of different light beams.
We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.
In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form $\mathfrak{f}$ with half-integral weight:\begin{itemize}\item[--]The first negative coefficient of the sequence $\{\mathfrak{a}\_{\mathfrak{f}}(tn^2)\}\_{n\in \N}$,\item[--]The number of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ of same signs,\item[--]Non-vanishing of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ in short intervals and in arithmetic progressions,\end{itemize}where $\mathfrak{a}\_{\mathfrak{f}}(n)$ is the $n$-th Fourier coefficient of $\mathfrak{f}$ and $t$ is a square-free integersuch that $\mathfrak{a}\_{\mathfrak{f}}(t)\not=0$.
In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implies a simple proof of the equivalence between the ADM mass and the intrinsically defined mass via the Ricci tensor, which was reconsidered by Miao-Tam \cite{MT} and Herzlich \cite{H} very recently.
In 1974, Landis and Oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time, then the solution must vanish identically before that time, provided the coefficients of the equation satisfy appropriate conditions at infinity. We prove this conjecture under some reasonable assumptions on the coefficients which improved the earlier results.
Let $π$ be an irreducible unitary cuspidal representation for $GL_m({\Bbb A}_{\Bbb Q})$, and let $L(s, π)$ be the automorphic $L$-function attached to $π$, which has a Dirichlet series expression in the half-plane $\mbox{Re} s>1$. When $π$ is self-contragredient, all the coefficients in the Dirichlet series expression are real. In this paper we give non-trivial lower bounds for the number of positive and negative coefficients, respectively.
For polarimetric SAR (PolSAR) image classification, it is a challenge to classify the aggregated terrain types, such as the urban area, into semantic homogenous regions due to sharp bright-dark variations in intensity. The aggregated terrain type is formulated by the similar ground objects aggregated together. In this paper, a polarimetric hierarchical semantic model (PHSM) is firstly proposed to overcome this disadvantage based on the constructions of a primal-level and a middle-level semantic. The primal-level semantic is a polarimetric sketch map which consists of sketch segments as the sparse representation of a PolSAR image. The middle-level semantic is a region map which can extract semantic homogenous regions from the sketch map by exploiting the topological structure of sketch segments. Mapping the region map to the PolSAR image, a complex PolSAR scene is partitioned into aggregated, structural and homogenous pixel-level subspaces with the characteristics of relatively coherent terrain types in each subspace. Then, according to the characteristics of three subspaces above, three specific methods are adopted, and furthermore polarimetric information is exploited to improve the segmentation result. Experimental results on PolSAR data sets with different bands and sensors demonstrate that the proposed method is superior to the state-of-the-art methods in region homogeneity and edge preservation for terrain classification.
Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf t\|$ for corresponding ideals $\bf r,\bf s, \bf t$ in the integral group ring $\mathbb Z[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\|\bf r,\bf s,\bf t\|)$ has exponent 2 modulo $\|R,S,T\|.$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.