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Jie Wu

Publications and source records attributed to Jie Wu.

At least 325 records · Page 18Linked to original sources
arXiv2013

A new characterization of geodesic spheres in the Hyperbolic space

This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a ``weighted'' higher order mean curvature. Precisely, we show that a compact hypersurface $Σ^{n-1}$ embedded in $\H^n$ with $VH_k$ being constant for some $k=1,\cdots,n-1$ is a centered geodesic sphere. Here $H_k$ is the $k$-th normalized mean curvature of $Σ$ induced from $\H^n$ and $V=\cosh r$, where $r$ is a hyperbolic distance to a fixed point in $\H^n$. Moreover, this result can be generalized to a compact hypersurface $Σ$ embedded in $\H^n$ with the ratio $V\left(\frac{H_k}{H_j}\right)\equiv\mbox{constant},\;0\leq j< k\leq n-1$ and $H_j$ not vanishing on $Σ$.

math.DG↗
arXiv2013

A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.

math.DG↗
arXiv2013

A positive mass theorem in the Einstein-Gauss-Bonnet theory

As an interesting application of the Einstein-Gauss-Bonnet theory and our work on the Gauss-Bonnet-Chern mass (Ge, Wang, Wu), we obtain a positive mass theorem for asymptotically flat graphs in $\R^{n+1}$ under a condition that $R+αL_2$ is non-negative, where $R$ is the scalar curvature, $α\in\R$ a constant and $L_2$ the second Gauss-Bonnet curvature. A Penrose type inequality is also obtained in the case $α>0$.

math.DG↗
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space $\H^n$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{equation*} \begin{array}{rcl} \ds \int_Σ\s_{2k}\ge \ds\vs C_{n-1}^{2k}ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 1k + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} \end{equation*} provided that $Σ$ is a horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $σ_{j}=\s_{j}(κ)$ is the $j$-th mean curvature and $κ=(κ_1,κ_2,\cdots, κ_{n-1})$ is the set of the principal curvatures of $Σ$. Also, an optimal inequality for quermassintegrals in $\H^n$ is as following: $$ W_{2k+1}(Ω)\geq\frac {ω_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(Ω)}{ω_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}}, $$ provided that $Ω\subset\H^n$ is a domain with $Σ=\partialΩ$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $W_r(Ω)$ is quermassintegrals in integral geometry.

math.DG↗
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_Σ\s_4 d μ\ge \ds\vs C_{n-1}^4ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 12 + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array}\] provided that $Σ$ is a horospherical convex hypersurface. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$.

math.DG↗
arXiv2013

Simplicial Monoid Actions and The Associated Universal Simplicial Monoid Construction

The reduced universal monoid on the action category associated to a pointed simplicial M-set has appeared in the guise of various simplicial monoid and group constructions. These include the classical constructions of Milnor and James, as well as their later generalizations by Carlsson and Wu. We prove that, if any two n-simplices in the same orbit differ by the action of an invertible monoid element, then the classifying space of this reduced universal monoid is the homotopy cofiber of the inclusion from the pointed simplicial set into its reduced Borel construction. The known formulae for the respective classifying spaces of the above four constructions are special cases of this result. Thus, we unify categorially the four above-mentioned constructions.

math.AT↗
arXiv2013

Evolution of superconducting correlations within magnetic-field-decoupled CuO(2) layers of La(1.905)Ba(0.095)CuO(4)

We explore the evolution of superconductivity in La(2-x)Ba(x)CuO(4) with x=0.095 in magnetic fields of up to 35 T applied perpendicular to the CuO(2) planes. Previous work on this material has shown that perpendicular fields enhance both charge and spin stripe order within the planes. We present measurements of the resistivity parallel and perpendicular to the planes, as well as the Hall effect. Measurements of magnetic susceptibility for fields of up to 15 T applied both parallel and perpendicular to the planes provide complementary measures of the superconductivity. We show that fields sufficient to destroy pair tunneling between the planes do not disrupt the superconducting correlations within the planes. In fact, we observe an onset of large amplitude but phase disordered superconductivity within the planes at approximately 30 K that is remarkably insensitive to field. With further cooling, we observe a phase-transition-like drop in the in-plane resistivity to an apparent state of superconductivity, despite the lack of phase coherence between the layers. These observations raise interesting questions concerning the identification of the upper critical field, where pairing is destroyed, in underdoped cuprates.

cond-mat.supr-con↗
arXiv2013

Homology Decompositions of the Loops on 1-Stunted Borel Constructions of C_2-Actions

Carlsson's construction is a simplicial group whose geometric realization is the loop space of the 1-stunted reduced Borel construction. Our main results are: i) Given a pointed simplicial set acted upon by the discrete cyclic group C_2 of order 2, if the orbit projection has a section, then this loop space has a mod 2 homology decomposition; ii) If the reduced diagonal map of the C_2-invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the C_2-invariant set and the orbit space. Result i) generalizes a previous homology decomposition of the second author for trivial actions. To illustrate these two results, we completely compute the mod 2 Betti numbers for an example.

math.AT↗
arXiv2012

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the $h$-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of Euler characteristic for the classical configuration space, which generalizes the Félix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retract of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which turns out that we are able to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

math.AT↗
arXiv2012

DQSB: A Reliable Broadcast Protocol Based on Distributed Quasi-Synchronized Mechanism for Low Duty-Cycled Wireless Sensor Networks

In duty-cycled wireless sensor networks, deployed sensor nodes are usually put to sleep for energy efficiency according to sleep scheduling approaches. Any sleep scheduling scheme with its supporting protocols ensures that data can always be routed from source to sink. In this paper, we investigate a problem of multi-hop broadcast and routing in random sleep scheduling scheme, and propose a novel protocol, called DQSB, by quasi-synchronization mechanism to achieve reliable broadcast and less latency routing. DQSB neither assumes time synchronization which requires all neighboring nodes wake up at the same time, nor assumes duty-cycled awareness which makes it difficult to use in asynchronous WSNs. Furthermore, the benefit of quasi-synchronized mechanism for broadcast from sink to other nodes is the less latency routing paths for reverse data collection to sink because of no or less sleep waiting time. Simulation results show that DQSB outperforms the existing protocols in broadcast times performance and keeps relative tolerant broadcast latency performance, even in the case of unreliable links. The proposed DQSB protocol, in this paper, can be recognized as a tradeoff between broadcast times and broadcast latency. We also explore the impact of parameters in the assumption and the approach to get proper values for supporting DQSB.

cs.NI↗
arXiv2012

The average exponent of elliptic curves modulo $p$

Let $E$ be an elliptic curve defined over ${\mathbb Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0, 1)$ such that $$ \frac{1}{π(x)} \sum_{p\le x} e_p = 1/2 C_E x + O_E\big(x^{5/6} (\log x)^{4/3}\big) $$ for all $x\ge 2$, where the implied constant depends on $E$ at most. When $E$ has complex multiplication, the same asymptotic formula with a weaker error term $O_E(1/(\log x)^{1/14})$ is established unconditionally. These improve some recent results of Freiberg and Kurlberg.

math.NT↗
arXiv2012

Modular representations and the homotopy of low rank $p$-local $CW$-complexes

Fix an odd prime $p$ and let $X$ be the $p$-localization of a finite suspended $CW$-complex. Given certain conditions on the reduced mod-$p$ homology $\bar H_*(X;\zmodp)$ of $X$, we use a decomposition of $ΩΣX$ due to the second author and computations in modular representation theory to show there are arbitrarily large integers $i$ such that $ΩΣ^i X$ is a homotopy retract of $ΩΣX$. This implies the stable homotopy groups of $ΣX$ are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for $ΣX$. Under additional assumptions on $\bar H_*(X;\zmodp)$, we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of $ΩΣX$ that has infinitely many finite $H$-spaces as factors.

math.AT↗
arXiv2011

On the general additive divisor problem

We obtain a new upper bound for $\sum_{h\le H}Δ_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $Δ_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $ζ(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.

math.NT↗
arXiv2011

A combinatorial description of homotopy groups of spheres

We give a combinatorial description of general homotopy groups of $k$-dimensional spheres with $k\geq3$ as well as those of Moore spaces. For $n>k\geq 3,$ we construct a finitely generated group defined by explicit generators and relations, whose center is exactly $π_n(S^k)$.

math.AT↗
arXiv2011

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in \cite{GL} \cite{Lee} \cite{Bi} of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.

math.DG↗