On the distribution of Jacobi sums

Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(\chi_1,\chi_2)$ ($\chi_1\chi_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $\chi_1$ and $\chi_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $\chi_1,\dots, \chi_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$ with two of the sets being sufficiently large. The case $m=2$ answers a question of Shparlinski.Comment: 18 pages. v3: fixed some typos; v2: improved some bound

Soft-Defined Heterogeneous Vehicular Network: Architecture and Challenges

Heterogeneous Vehicular NETworks (HetVNETs) can meet various quality-of-service (QoS) requirements for intelligent transport system (ITS) services by integrating different access networks coherently. However, the current network architecture for HetVNET cannot efficiently deal with the increasing demands of rapidly changing network landscape. Thanks to the centralization and flexibility of the cloud radio access network (Cloud-RAN), soft-defined networking (SDN) can conveniently be applied to support the dynamic nature of future HetVNET functions and various applications while reducing the operating costs. In this paper, we first propose the multi-layer Cloud RAN architecture for implementing the new network, where the multi-domain resources can be exploited as needed for vehicle users. Then, the high-level design of soft-defined HetVNET is presented in detail. Finally, we briefly discuss key challenges and solutions for this new network, corroborating its feasibility in the emerging fifth-generation (5G) era

Multi-center clinical trials: Randomization and ancillary statistics

The purpose of this paper is to investigate and develop methods for analysis of multi-center randomized clinical trials which only rely on the randomization process as a basis of inference. Our motivation is prompted by the fact that most current statistical procedures used in the analysis of randomized multi-center studies are model based. The randomization feature of the trials is usually ignored. An important characteristic of model based analysis is that it is straightforward to model covariates. Nevertheless, in nearly all model based analyses, the effects due to different centers and, in general, the design of the clinical trials are ignored. An alternative to a model based analysis is to have analyses guided by the design of the trial. Our development of design based methods allows the incorporation of centers as well as other features of the trial design. The methods make use of conditioning on the ancillary statistics in the sample space generated by the randomization process. We have investigated the power of the methods and have found that, in the presence of center variation, there is a significant increase in power. The methods have been extended to group sequential trials with similar increases in power.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS151 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Categorical traces and a relative Lefschetz-Verdier formula

We prove a relative Lefschetz-Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$-category of cohomological correspondences. We show that local acyclicity is equivalent to dualizability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalizing a theorem of Gabber.Comment: 26 pages. v3: minor improvement

Conformal Ricci flow on asymptotically hyperbolic manifolds

In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.Comment: 19 pages. No figur