Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kaehler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.Comment: 71 pages; v.2 is a final update to agree with the published pape

On the stability of equivariant embedding of compact CR manifolds with circle action

We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.Comment: 21 pages, final versio

Auction-Based Coopetition between LTE Unlicensed and Wi-Fi

Motivated by the recent efforts in extending LTE to the unlicensed spectrum, we propose a novel spectrum sharing framework for the coopetition (i.e., cooperation and competition) between LTE and Wi-Fi in the unlicensed band. Basically, the LTE network can choose to work in one of the two modes: in the competition mode, it randomly accesses an unlicensed channel, and interferes with the Wi-Fi access point using the same channel; in the cooperation mode, it delivers traffic for the Wi-Fi users in exchange for the exclusive access of the corresponding channel. Because the LTE network works in an interference-free manner in the cooperation mode, it can achieve a much larger data rate than that in the competition mode, which allows it to effectively serve both its own users and the Wi-Fi users. We design a second-price reverse auction mechanism, which enables the LTE provider and the Wi-Fi access point owners (APOs) to effectively negotiate the operation mode. Specifically, the LTE provider is the auctioneer (buyer), and the APOs are the bidders (sellers) who compete to sell their channel access opportunities to the LTE provider. In Stage I of the auction, the LTE provider announces a reserve rate. In Stage II of the auction, the APOs submit their bids. We show that the auction involves allocative externalities, i.e., the cooperation between the LTE provider and one APO benefits other APOs who are not directly involved in this cooperation. As a result, a particular APO's willingness to cooperate is affected by its belief about other APOs' willingness to cooperate. This makes our analysis much more challenging than that of the conventional second-price auction, where bidding truthfully is a weakly dominant strategy. We show that the APOs have a unique form of the equilibrium bidding strategies in Stage II, based on which we analyze the LTE provider's optimal reserve rate in Stage I.Comment: 32 page

Geometry-Oblivious FMM for Compressing Dense SPD Matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in $N \log N$ or even $N$ time, where $N$ is the matrix size. Compression requires $N \log N$ storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1