Compactifications of Complete Riemannian manifolds and Their Applications
To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.
math.DG↗
arXiv subjects
Xiaodong Wang
Publications and source records attributed to Xiaodong Wang.
Bottom of spectrum of Kahler manifolds with strongly pseudoconvex boundary
We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.
math.DG↗
Local Gradient Estimate for $p$-harmonic functions on Riemannian Manifolds
For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on which the function is defined. Our approach is based on a careful application of the Moser iteration technique and is different from Cheng-Yau's method employed by Kostchwar and Ni, in which a gradient estimate for positive $p$-harmonic functions is derived under the assumption that the sectional curvature is bounded from below.
math.DG↗
Robust Linear Precoder Design for Multi-cell Downlink Transmission
Coordinated information processing by the base stations of multi-cell wireless networks enhances the overall quality of communication in the network. Such coordinations for optimizing any desired network-wide quality of service (QoS) necessitate the base stations to acquire and share some channel state information (CSI). With perfect knowledge of channel states, the base stations can adjust their transmissions for achieving a network-wise QoS optimality. In practice, however, the CSI can be obtained only imperfectly. As a result, due to the uncertainties involved, the network is not guaranteed to benefit from a globally optimal QoS. Nevertheless, if the channel estimation perturbations are confined within bounded regions, the QoS measure will also lie within a bounded region. Therefore, by exploiting the notion of robustness in the worst-case sense some worst-case QoS guarantees for the network can be asserted. We adopt a popular model for noisy channel estimates that assumes that estimation noise terms lie within known hyper-spheres. We aim to design linear transceivers that optimize a worst-case QoS measure in downlink transmissions. In particular, we focus on maximizing the worst-case weighted sum-rate of the network and the minimum worst-case rate of the network. For obtaining such transceiver designs, we offer several centralized (fully cooperative) and distributed (limited cooperation) algorithms which entail different levels of complexity and information exchange among the base stations.
cs.IT↗
Effect of annealing temperature on morphology, structure and photocatalytic behavior of nanotubed H2Ti2O4(OH)2
Nanotubed titanic acid (H2Ti2O4(OH)2) was prepared from nanotubed sodium titanate (Na2Ti2O4(OH)2) by an ion exchange reaction in a pH=1 HCl solution. The effect of annealing temperature on the morphology, structure and photocatalytic behavior of nanotubed H2Ti2O4(OH)2 was studied by means of TEM, XRD, DTG, DSC, BET and ESR. The results showed that nanotubed H2Ti2O4(OH)2 is thermally unstable. Its dehydration consists of two steps. In the first-step dehydration, single-electron-trapped oxygen vacancies (SETOVs) were generated. Accompanying the second-step dehydration, the transition of crystal form from orthorhombic system to anatase took place, at the same time the nanotubes broke. At T>300 °C, when the SETOV concentration greatly increased, the interaction between SETOV happened. (VOo)x formed could play the role of recombination center of photogenerated e--h+ and make the photocatalytic behavior of TiO2 (anatase, obtained from 500 °C-treated nanotubed H2Ti2O4(OH)2) to become bad.
cond-mat.mtrl-sci↗
Superimposed XOR: Approaching Capacity Bounds of the Two-Way Relay Channels
In two-way relay channels, bitwise XOR and symbol-level superposition coding are two popular network-coding based relaying schemes. However, neither of them can approach the capacity bound when the channels in the broadcast phase are asymmetric. In this paper, we present a new physical layer network coding (PLNC) scheme, called \emph{superimposed XOR}. The new scheme advances the existing schemes by specifically taking into account the channel asymmetry as well as information asymmetry in the broadcast phase. We obtain its achievable rate regions over Gaussian channels when integrated with two known time control protocols in two-way relaying. We also demonstrate their average maximum sum-rates and service delay performances over fading channels. Numerical results show that the proposed superimposed XOR achieves a larger rate region than both XOR and superposition and performs much better over fading channels. We further deduce the boundary of its achievable rate region of the broadcast phase in an explicit and analytical expression. Based on these results, we then show that the gap to the capacity bound approaches zero at high signal-to-noise ratio.
cs.IT↗
(n,K)-user Interference Channels: Degrees of Freedom
We analyze the gains of opportunistic communication in multiuser interference channels. Consider a fully connected $n$-user Gaussian interference channel. At each time instance only $K\leq n$ transmitters are allowed to be communicating with their respective receivers and the remaining $(n-K)$ transmitter-receiver pairs remain inactive. For finite $n$, if the transmitters can acquire channel state information (CSI) and if all channel gains are bounded away from zero and infinity, the seminal results on interference alignment establish that for any $K$ {\em arbitrary} active pairs the total number of spatial degrees of freedom per orthogonal time and frequency domain is $\frac{K}{2}$. Also it is noteworthy that without transmit-side CSI the interference channel becomes interference-limited and the degrees of freedom is 0. In {\em dense} networks ($n\rightarrow\infty$), however, as the size of the network increase, it becomes less likely to sustain the bounding conditions on the channel gains. By exploiting this fact, we show that when $n$ obeys certain scaling laws, by {\em opportunistically} and {\em dynamically} selecting the $K$ active pairs at each time instance, the number of degrees of freedom can exceed $\frac{K}{2}$ and in fact can be made arbitrarily close to $K$. More specifically when all transmitters and receivers are equipped with one antenna, then the network size scaling as $n\inω(\snr^{d(K-1)})$ is a {\em sufficient} condition for achieving $d\in[0,K]$ degrees of freedom. Moreover, achieving these degrees of freedom does not necessitate the transmitters to acquire channel state information. Hence, invoking opportunistic communication in the context of interference channels leads to achieving higher degrees of freedom that are not achievable otherwise.
cs.IT↗
Multiuser Diversity Gain in Cognitive Networks
Dynamic allocation of resources to the \emph{best} link in large multiuser networks offers considerable improvement in spectral efficiency. This gain, often referred to as \emph{multiuser diversity gain}, can be cast as double-logarithmic growth of the network throughput with the number of users. In this paper we consider large cognitive networks granted concurrent spectrum access with license-holding users. The primary network affords to share its under-utilized spectrum bands with the secondary users. We assess the optimal multiuser diversity gain in the cognitive networks by quantifying how the sum-rate throughput of the network scales with the number of secondary users. For this purpose we look at the optimal pairing of spectrum bands and secondary users, which is supervised by a central entity fully aware of the instantaneous channel conditions, and show that the throughput of the cognitive network scales double-logarithmically with the number of secondary users ($N$) and linearly with the number of available spectrum bands ($M$), i.e., $M\log\log N$. We then propose a \emph{distributed} spectrum allocation scheme, which does not necessitate a central controller or any information exchange between different secondary users and still obeys the optimal throughput scaling law. This scheme requires that \emph{some} secondary transmitter-receiver pairs exchange $\log M$ information bits among themselves. We also show that the aggregate amount of information exchange between secondary transmitter-receiver pairs is {\em asymptotically} equal to $M\log M$. Finally, we show that our distributed scheme guarantees fairness among the secondary users, meaning that they are equally likely to get access to an available spectrum band.
cs.IT↗
An integral formula for the volume entropy with applications to rigidity
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
math.DG↗
Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature
We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.
math.DG↗
Optimal Joint Target Detection and Parameter Estimation By MIMO Radar
We consider multiple-input multiple-output (MIMO) radar systems with widely-spaced antennas. Such antenna configuration facilitates capturing the inherent diversity gain due to independent signal dispersion by the target scatterers. We consider a new MIMO radar framework for detecting a target that lies in an unknown location. This is in contrast with conventional MIMO radars which break the space into small cells and aim at detecting the presence of a target in a specified cell. We treat this problem through offering a novel composite hypothesis testing framework for target detection when (i) one or more parameters of the target are unknown and we are interested in estimating them, and (ii) only a finite number of observations are available. The test offered optimizes a metric which accounts for both detection and estimation accuracies. In this paper as the parameter of interest we focus on the vector of time-delays that the waveforms undergo from being emitted by the transmit antennas until being observed by the receive antennas. The analytical and empirical results establish that for the proposed joint target detection and time-delay estimation framework, MIMO radars exhibit significant gains over phased-array radars for extended targets which consist of multiple independent scatterers. For point targets modeled as single scatterers, however, the detection/estimation accuracies of MIMO and phased-array radars for this specific setup (joint target detection and time-delay estimation) are comparable.
cs.IT↗
Beamforming and Rate Allocation in MISO Cognitive Radio Networks
We consider decentralized multi-antenna cognitive radio networks where secondary (cognitive) users are granted simultaneous spectrum access along with license-holding (primary) users. We treat the problem of distributed beamforming and rate allocation for the secondary users such that the minimum weighted secondary rate is maximized. Such an optimization is subject to (1) a limited weighted sum-power budget for the secondary users and (2) guaranteed protection for the primary users in the sense that the interference level imposed on each primary receiver does not exceed a specified level. Based on the decoding method deployed by the secondary receivers, we consider three scenarios for solving this problem. In the first scenario each secondary receiver decodes only its designated transmitter while suppressing the rest as Gaussian interferers (single-user decoding). In the second case each secondary receiver employs the maximum likelihood decoder (MLD) to jointly decode all secondary transmissions, and in the third one each secondary receiver uses the unconstrained group decoder (UGD). By deploying the UGD, each secondary user is allowed to decode any arbitrary subset of users (which contains its designated user) after suppressing or canceling the remaining users.
cs.IT↗
Beacon-Assisted Spectrum Access with Cooperative Cognitive Transmitter and Receiver
Spectrum access is an important function of cognitive radios for detecting and utilizing spectrum holes without interfering with the legacy systems. In this paper we propose novel cooperative communication models and show how deploying such cooperations between a pair of secondary transmitter and receiver assists them in identifying spectrum opportunities more reliably. These cooperations are facilitated by dynamically and opportunistically assigning one of the secondary users as a relay to assist the other one which results in more efficient spectrum hole detection. Also, we investigate the impact of erroneous detection of spectrum holes and thereof missing communication opportunities on the capacity of the secondary channel. The capacity of the secondary users with interference-avoiding spectrum access is affected by 1) how effectively the availability of vacant spectrum is sensed by the secondary transmitter-receiver pair, and 2) how correlated are the perceptions of the secondary transmitter-receiver pair about network spectral activity. We show that both factors are improved by using the proposed cooperative protocols. One of the proposed protocols requires explicit information exchange in the network. Such information exchange in practice is prone to wireless channel errors (i.e., is imperfect) and costs bandwidth loss. We analyze the effects of such imperfect information exchange on the capacity as well as the effect of bandwidth cost on the achievable throughput. The protocols are also extended to multiuser secondary networks.
cs.IT↗
Design of Learning Based MIMO Cognitive Radio Systems
This paper addresses the design issues of the multi-antenna-based cognitive radio (CR) system that is able to operate concurrently with the licensed primary radio (PR) system. We propose a practical CR transmission strategy consisting of three major stages: environment learning, channel training, and data transmission. In the environment learning stage, the CR transceivers both listen to the PR transmission and apply blind algorithms to estimate the spaces that are orthogonal to the channels from the PR. Assuming time-division duplex (TDD) based transmission for the PR, cognitive beamforming is then designed and applied at CR transceivers to restrict the interference to/from the PR during the subsequent channel training and data transmission stages. In the channel training stage, the CR transmitter sends training signals to the CR receiver, which applies the linear-minimum-mean-square-error (LMMSE) based estimator to estimate the effective channel. Considering imperfect estimations in both learning and training stages, we derive a lower bound on the ergodic capacity achievable for the CR in the data transmission stage. From this capacity lower bound, we observe a general learning/training/throughput tradeoff associated with the proposed scheme, pertinent to transmit power allocation between training and transmission stages, as well as time allocation among learning, training, and transmission stages. We characterize the aforementioned tradeoff by optimizing the associated power and time allocation to maximize the CR ergodic capacity.
cs.IT↗
Ricci and Scalar Curvature Rigidity of the Hemisphere
We retract the scalar curvature rigidity theorem as there is a mistake in the proof. We thank S. Montiel for pointing out the mistake.
math.DG↗
A Rigidity Theorem for the Hemisphere
We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally geodesic, then the manifold is isometric to the standard hemisphere.
math.DG↗
Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space
Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is well known that the bottom of spectrum $λ_{0}$ of its unverversal covering satisfies $λ_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.
math.DG↗
Deconfinement Phase Transition Heating and Thermal Evolution of Neutron Stars
The deconfinement phase transition will lead to the release of latent heat during spins down of neutron stars if the transition is the first-order one.We have investigated the thermal evolution of neutron stars undergoing such deconfinement phase transition. The results show that neutron stars may be heated to higher temperature.This feature could be particularly interesting for high temperature of low-magnetic field millisecond pulsar at late stage.
astro-ph↗