arXiv subjects
Publications and source records attributed to Jian Song.
The stable reduction theorem says that a family of curves of genus $g\geq 2$ over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new proof of this result for curves defined over $\mathbb{C}$ using the Kähler-Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.
Let $\mathcal{M}_{\rm KSB}$ (resp. $\mathcal{M}_{\rm KSB}'$) be the the moduli space of $n$-dimensional Kähler-Einstein manifolds (resp. varieties) $X$ with $K_X$ ample. We prove that the Weil-Petersson metric on $\mathcal{M}_{\rm KSB}$ extends uniquely to the projective variety $\mathcal{M}_{\rm KSB}'$, as a closed positive current with continuous local potentials. This generalizes a theorem of Wolpert which treats the case $n=1$, and also confirms a conjecture of Berman-Guenancia. In addition, we derive uniform estimates for the volumes of sublevel sets of Kähler-Einstein potentials
In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t), t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in(1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of $\{L_N(t), t\in[0,T]\}_{N\in\mathbb N}$ in $C([0,T], \mathbf P(\mathbb R))$ following the approach in \cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of $\{L_N(t), t\in[0,T]\}_{N\in\mathbb N}$ on $C([0,T], \mathbf P(\mathbb R))$ by tightness criterions provided in Appendix \ref{subset:tightness argument}. The limit of $\{L_N(t), t\in[0,T]\}$ as $N\to \infty$ is also characterised.
Unveiling a fundamental link between information theory and estimation theory, the I-MMSE relationship by Guo, Shamai and Verdu~\cite{gu05}, together with its numerous extensions, has great theoretical significance and various practical applications. On the other hand, its influences to date have been restricted to channels without feedback or memory, due to the absence of its extensions to such channels. In this paper, we propose extensions of the I-MMSE relationship to discrete-time and continuous-time Gaussian channels with feedback and/or memory. Our approach is based on a very simple observation, which can be applied to other scenarios, such as a simple and direct proof of the classical de Bruijn's identity. This submission corrects the mistakes in the previous version.
For real symmetric and complex Hermitian Gaussian processes whose values are $d\times d$ matrices, we characterize the conditions under which the probability that at least $k$ eigenvalues collide is positive for $2\le k\le d$, and we obtain the Hausdorff dimension of the set of collision times.
This work is motivated by the need of collecting fresh data from power-constrained sensors in the industrial Internet of Things (IIoT) network. A recently proposed metric, the Age of Information (AoI) is adopted to measure data freshness from the perspective of the central controller in the IIoT network. We wonder what is the minimum average AoI the network can achieve and how to design scheduling algorithms to approach it. To answer these questions when the channel states of the network are Markov time-varying and scheduling decisions are restricted to bandwidth constraint, we first decouple the multi-sensor scheduling problem into a single-sensor constrained Markov decision process (CMDP) through relaxation of the hard bandwidth constraint. Next we exploit the threshold structure of the optimal policy for the decoupled single sensor CMDP and obtain the optimum solution through linear programming (LP). Finally, an asymptotically optimal truncated policy that can satisfy the hard bandwidth constraint is built upon the optimal solution to each of the decoupled single-sensor. Our investigation shows that to obtain a small AoI performance: (1) The scheduler exploits good channels to schedule sensors supported by limited power; (2) Sensors equipped with enough transmission power are updated in a timely manner such that the bandwidth constraint can be satisfied.
Let $\mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence $\{ (X_j, g_j)\}_{j=1}^\infty$ in $\mathcal{K}(n, V)$ converges, in the pointed Gromov-Hausdorff topology, to a finite union of complete Kahler-Einstein metric spaces without loss of volume. The convergence is smooth off a closed singular set of Hausdorff dimension no greater than $2n-4$, and the limiting metric space is biholomorphic to an $n$-dimensional semi-log canonical model with its non log terminal locus of complex dimension no greater than $n-1$ removed. We also show that the Weil-Petersson metric extends uniquely to a Kahler current with bounded local potentials on the KSBA compactification of the moduli space of canonically polarized manifolds. In particular, the coarse KSBA moduli space has finite volume with respect to the Weil-Petersson metric. Our results are a high dimensional generalization of the well known compactness results for hyperbolic metrics on compact Riemann surfaces of fixed genus greater than one.
In this work, a wireless broadcast network with a base station (BS) sending random time-sensitive information updates to multiple users with interference constraints is considered. The Age of Synchronization (AoS), namely the amount of time elapsed since the information stored at the network user becomes desynchronized, is adopted to measure data freshness from the perspective of network users. Compared with the more widely used metric---the Age of Information (AoI), AoS accounts for the freshness of the randomly changing content. The AoS minimization scheduling problem is formulated into a discrete time Markov decision process and the optimal solution is approximated through structural finite state policy iteration. An index based heuristic scheduling policy based on restless multi-arm bandit (RMAB) is provided to further reduce computational complexity. Simulation results show that the proposed index policy can achieve compatible performance with the MDP and close to the AoS lower bound. Moreover, theoretic analysis and simulations reveal the differences between AoS and AoI. AoI minimization scheduling policy cannot guarantee a good AoS performance.
Motivated by the classical De Bruijn's identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in(0,1)$. We derive generalized De Bruijn's identity for Shannon entropy and Kullback-Leibler divergence by means of Itô's formula, and present two applications. In the first application we demonstrate its equivalence with Stein's identity for Gaussian distributions, while in the second application, we show that for $H \in (0,1/2]$, the entropy power is concave in time while for $H \in (1/2,1)$ it is convex in time when the initial distribution is Gaussian. Compared with the classical case of $H = 1/2$, the time parameter plays an interesting and significant role in the analysis of these quantities.
In this paper, we study how to collect fresh data in time-varying networks with power constrained users. We measure data freshness from the perspective of the central controller by using the metric Age of Information, namely the time elapsed since the generation time-stamp of the freshest information. We wonder what is the minimum AoI performance the network can achieve and how to design scheduling algorithms to approach it. To answer these questions when scheduling decisions are restricted to bandwidth constraint, we first decouple the multi-user scheduling problem into a single user constrained Markov decision process (CMDP) through relaxation of the hard bandwidth constraint. Next we exploit the threshold structure of the optimal policy for the decoupled single user CMDP and obtain the optimum solution through linear programming (LP). Finally, an asymptotic optimal truncated policy that can satisfy the hard bandwidth constraint is built upon the optimal solution to each of the decoupled single-user sub-problem. The performance is verified through simulations. Our investigation shows that to obtain a small AoI performance, the scheduler exploits good channels to schedule users supported by limited power. Users equipped with enough transmission power are updated in a timely manner such that the bandwidth constraint can be satisfied.
We consider a scenario where a base station (BS) attempts to collect fresh information from power constrained sensors over time-varying band-limited wireless channels. We characterize the data freshness through the recently proposed metric--the Age of Information. We consider a time-varying channel model with power adaptation. Unlike previous work, packet loss may happen due to imperfect channel estimation or decoding error. We propose an asymptotic optimal scheduling algorithm minimizing AoI performance and satisfying both bandwidth and power constraint in such networks. Numerical simulations show that the proposed policy outperforms the greedy one, and we observe that sensors with poor channels are scheduled at higher AoI in order to limit the packet loss.
In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.
The research about deep learning application for physical layer has been received much attention in recent years. In this paper, we propose a Deep Learning (DL) based channel estimator under time varying Rayleigh fading channel. We build up, train and test the channel estimator using Neural Network (NN). The proposed DL-based estimator can dynamically track the channel status without any prior knowledge about the channel model and statistic characteristics. The simulation results show the proposed NN estimator has better Mean Square Error (MSE) performance compared with the traditional algorithms and some other DL-based architectures. Furthermore, the proposed DL-based estimator also shows its robustness with the different pilot densities.
We consider eigenvalues of generalized Wishart processes as well as particle systems, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, central limit theorems for the Dyson's Brownian motion and the eigenvalues of the Wishart process are recovered under slightly more general initial conditions, and a central limit theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process is obtained.
Solutions of manually purified gastric mucins have been shown to be promising lubricants for biomedical purposes, where they can efficiently reduce friction and wear. However, so far, such mucin solutions have been mostly tested in specific settings, and variations in the composition of the lubricating fluid have not been systematically explored. We here fill this gap and determine the viscosity, adsorption behavior, and lubricity of porcine gastric mucin solutions on hydrophobic surfaces at different pH levels, mucin and salt concentrations and in the presence of other proteins. We demonstrate that mucin solutions provide excellent lubricity even at very low concentrations of 0.01 % (w/v), over a broad range of pH levels and even at elevated ionic strength. Furthermore, we provide mechanistic insights into mucin lubricity, which help explain how certain variations in physiologically relevant parameters can limit the lubricating potential of mucin solutions. Our results motivate that solutions of manually purified mucin solutions can be powerful biomedical lubricants, e.g. serving as eye drops, mouth sprays or as a personal lubricant for intercourse.
In this article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac14, \frac12)$ in space. We prove the existence and uniqueness of the mild Skorohod solution, establish lower and upper bounds for the $p$-th moment of the solution for all $p\ge2$, and obtain the Hölder continuity in time and space variables for the solution.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when $X$ has Kodaira dimension $1$ with $K_X$ being semi-ample and the general fibre of $X$ over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.
The combination of millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems and index modulation (IM) technique has recently constituted a novel form of mmWave hybrid beamforming. Lots of studies have been conducted to show that such system has the potential to outperform conventional mmWave-MIMOs with respect to spectral efficiency (SE). Most of the current works only focused on designing hybrid beamforming structures to empirically achieve higher SE performance. However, a fundamental question that whether IM technique can truly improve the SE of current mmWave hybrid beamforming is still left unanswered. Against such background, in this work we firstly extend the IM-assisted mmWave system to a more generalized version, i.e. a spatial path index modulation aided mmWave (SPIM-mmWave) system, which subsumes both IM-assisted and conventional mmWave-MIMO systems as its special cases. Based on the framework of SPIM-mmWave, a fundamental study is conducted towards a theoretic condition under which SPIM-mmWave guarantees to outperform conventional mmWave-MIMO schemes. It is demonstrated that, under a specific channel condition and noise level (which will be explicitly given by our work), SPIM-mmWave guarantees to outperform conventional mmWave-MIMO systems.