Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.Comment: 42 pages. Comments are welcom

Finite groups with small number of cyclic subgroups

In this note, we study the finite groups with the number of cylic subgroups no greater than 6.Comment: 4 page

Interior regularity of fully nonlinear degenerate elliptic equations, I: Bellman equations with constant coefficients

This is the first of a series of papers on the interior regularity of fully nonlinear degenerate elliptic equations. We consider a stochastic optimal control problem in which the diffusion coefficients, drift coefficients and discount factor are independent of the spacial variables. Under suitable assumptions, for $k=0,1$, when the terminal and running payoffs are globally $C^{k,1}$, we obtain the $C^{k,1}$-smoothness of the value function, which yields the existence and uniqueness of the solution to the associated Dirichlet problem for the degenerate Bellman equation.Comment: Assumption 2.2 was corrected and then weakened. The original Assumption 2.2 after correction is now Remark 2.1. Minor revision was made accordingly on pages 28 and 29. A few typos were corrected als

On representation and regularity of viscosity solutions to degenerate Isaacs equations and certain nonconvex Hessian equations

We study the smoothness of the upper and lower value functions of stochastic differential games in the framework of time-homogeneous (possibly degenerate) diffusion processes in a domain, under the assumption that the diffusion, drift and discount coefficients are all independent of the spatial variables. Under suitable conditions (see Assumptions 2.1 and 2.2), we obtain the optimal local Lipschitz continuity of the value functions, provided that the running and terminal payoffs are globally Lipschitz. As applications, we obtain the stochastic representation and optimal interior $C^{0,1}$-regularity of the unique viscosity solution to the Dirichlet problem for certain degenerate elliptic, nonconvex Hessian equations in suitable domains, with Lipschitz boundary data.Comment: 30 pages. Comments are welcom

Universal price impact functions of individual trades in an order-driven market

The trade size $\omega$ has direct impact on the price formation of the stock traded. Econophysical analyses of transaction data for the US and Australian stock markets have uncovered market-specific scaling laws, where a master curve of price impact can be obtained in each market when stock capitalization $C$ is included as an argument in the scaling relation. However, the rationale of introducing stock capitalization in the scaling is unclear and the anomalous negative correlation between price change $r$ and trade size $\omega$ for small trades is unexplained. Here we show that these issues can be addressed by taking into account the aggressiveness of orders that result in trades together with a proper normalization technique. Using order book data from the Chinese market, we show that trades from filled and partially filled limit orders have very different price impact. The price impact of trades from partially filled orders is constant when the volume is not too large, while that of filled orders shows power-law behavior $r\sim \omega^\alpha$ with $\alpha\approx2/3$. When returns and volumes are normalized by stock-dependent averages, capitalization-independent scaling laws emerge for both types of trades. However, no scaling relation in terms of stock capitalization can be constructed. In addition, the relation $\alpha=\alpha_\omega/\alpha_r$ is verified, where $\alpha_\omega$ and $\alpha_r$ are the tail exponents of trade sizes and returns. These observations also enable us to explain the anomalous negative correlation between $r$ and $\omega$ for small-size trades. We anticipate that these regularities may hold in other order-driven markets.Comment: 17 pages + supplementary figures. The paper has been significantly expanded and more Supplementary Information is adde

Quadboost: A Scalable Concurrent Quadtree

Building concurrent spatial trees is more complicated than binary search trees since a space hierarchy should be preserved during modifications. We present a non-blocking quadtree-quadboost-that supports concurrent insert, remove, move, and contain operations. To increase its concurrency, we propose a decoupling approach that separates physical adjustment from logical removal within the remove operation. In addition, we design a continuous find mechanism to reduce its search cost. The move operation combines the searches for different keys together and modifies different positions with atomicity. The experimental results show that quadboost scales well on a multi-core system with 32 hardware threads. More than that, it outperforms existing concurrent trees in retrieving two-dimensional keys with up to 109% improvement when the number of threads is large. The move operation proved to perform better than the best-known algorithm, with up to 47%

Uplink Multicell Processing with Limited Backhaul via Per-Base-Station Successive Interference Cancellation

This paper studies an uplink multicell joint processing model in which the base-stations are connected to a centralized processing server via rate-limited digital backhaul links. Unlike previous studies where the centralized processor jointly decodes all the source messages from all base-stations, this paper proposes a suboptimal achievability scheme in which the Wyner-Ziv compress-and-forward relaying technique is employed on a per-base-station basis, but successive interference cancellation (SIC) is used at the central processor to mitigate multicell interference. This results in an achievable rate region that is easily computable, in contrast to the joint processing schemes in which the rate regions can only be characterized by exponential number of rate constraints. Under the per-base-station SIC framework, this paper further studies the impact of the limited-capacity backhaul links on the achievable rates and establishes that in order to achieve to within constant number of bits to the maximal SIC rate with infinite-capacity backhaul, the backhaul capacity must scale logarithmically with the signal-to-interference-and-noise ratio (SINR) at each base-station. Finally, this paper studies the optimal backhaul rate allocation problem for an uplink multicell joint processing model with a total backhaul capacity constraint. The analysis reveals that the optimal strategy that maximizes the overall sum rate should also scale with the log of the SINR at each base-station.Comment: JSAC Oct 2013, special issue on VMIM

Capacity of the Gaussian Relay Channel with Correlated Noises to Within a Constant Gap

This paper studies the relaying strategies and the approximate capacity of the classic three-node Gaussian relay channel, but where the noises at the relay and at the destination are correlated. It is shown that the capacity of such a relay channel can be achieved to within a constant gap of \hf \log_2 3 =0.7925 bits using a modified version of the noisy network coding strategy, where the quantization level at the relay is set in a correlation dependent way. As a corollary, this result establishes that the conventional compress-and-forward scheme also achieves to within a constant gap to the capacity. In contrast, the decode-and-forward and the single-tap amplify-and-forward relaying strategies can have an infinite gap to capacity in the regime where the noises at the relay and at the destination are highly correlated, and the gain of the relay-to-destination link goes to infinity.Comment: accepted to communications letter

Optimized Backhaul Compression for Uplink Cloud Radio Access Network

This paper studies the uplink of a cloud radio access network (C-RAN) where the cell sites are connected to a cloud-computing-based central processor (CP) with noiseless backhaul links with finite capacities. We employ a simple compress-and-forward scheme in which the base-stations(BSs) quantize the received signals and send the quantized signals to the CP using either distributed Wyner-Ziv coding or single-user compression. The CP decodes the quantization codewords first, then decodes the user messages as if the remote users and the cloud center form a virtual multiple-access channel (VMAC). This paper formulates the problem of optimizing the quantization noise levels for weighted sum rate maximization under a sum backhaul capacity constraint. We propose an alternating convex optimization approach to find a local optimum solution to the problem efficiently, and more importantly, establish that setting the quantization noise levels to be proportional to the background noise levels is near optimal for sum-rate maximization when the signal-to-quantization-noise ratio (SQNR) is high. In addition, with Wyner-Ziv coding, the approximate quantization noise level is shown to achieve the sum-capacity of the uplink C-RAN model to within a constant gap. With single-user compression, a similar constant-gap result is obtained under a diagonal dominant channel condition. These results lead to an efficient algorithm for allocating the backhaul capacities in C-RAN. The performance of the proposed scheme is evaluated for practical multicell and heterogeneous networks. It is shown that multicell processing with optimized quantization noise levels across the BSs can significantly improve the performance of wireless cellular networks.Comment: 13 pages, 8 figures; published in IEEE Journal on Selected Areas in Communications, Special Issue on 5G Communication Systems, June 201

Normal property, Jamenson property, CHIP and linear regularity for an infinite system of convex sets in Banach spaces

In this paper, we study different kinds of normal properties for infinite system of arbitrarily many convex sets in a Banach space and provide the dual characterization for the normal property in terms of the extended Jamenson property for arbitrarily many weak*-closed convex cones in the dual space. Then, we use the normal property and the extended Jamenson property to study CHIP, strong CHIP and linear regularity for the infinite case of arbitrarily many convex sets and establish equivalent relationship among these properties. In particular, we extend main results in [3] on normal property, Jamenson property, CHIP and linear regularity for finite system of convex sets in a Hilbert space to the infinite case of arbitrarily many convex sets in Banach space setting