On certain properties of the pp-unitary Cayley graph over a finite ring

In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to be prime. In this paper, we continue this line of research. Specifically, we investigate some basic properties of certain $p$-unitary Cayeley graphs associated with a finite commutative ring. In particular, under some mild conditions, we provide the necessary and sufficient conditions for this graph to be prime.Comment: Comments are welcome

On the arithmetic of generalized Fekete polynomials

For each prime number $p$ one can associate a Fekete polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be a composite number. We then investigate the appearance of cyclotomic factors of these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.Comment: To appear in Experimental Mathematic

Rateless codes-based secure communication employing transmit antenna selection and harvest-to-jam under joint effect of interference and hardware impairments

In this paper, we propose a rateless codes-based communication protocol to provide security for wireless systems. In the proposed protocol, a source uses the transmit antenna selection (TAS) technique to transmit Fountain-encoded packets to a destination in presence of an eavesdropper. Moreover, a cooperative jammer node harvests energy from radio frequency (RF) signals of the source and the interference sources to generate jamming noises on the eavesdropper. The data transmission terminates as soon as the destination can receive a sufficient number of the encoded packets for decoding the original data of the source. To obtain secure communication, the destination must receive sufficient encoded packets before the eavesdropper. The combination of the TAS and harvest-to-jam techniques obtains the security and efficient energy via reducing the number of the data transmission, increasing the quality of the data channel, decreasing the quality of the eavesdropping channel, and supporting the energy for the jammer. The main contribution of this paper is to derive exact closed-form expressions of outage probability (OP), probability of successful and secure communication (SS), intercept probability (IP) and average number of time slots used by the source over Rayleigh fading channel under the joint impact of co-channel interference and hardware impairments. Then, Monte Carlo simulations are presented to verify the theoretical results.Web of Science217art. no. 70

On the Paley graph of a quadratic character

Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number $p$ we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo $p$. Therefore, Paley graphs are naturally associated with the Legendre symbol at $p$ which is a quadratic Dirichlet character of conductor $p$. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of $L$-functions, we provide an effective upper bound for their Cheeger number.Comment: To appear in Mathematica Slovac

Energy-Efficient Design for Downlink Cloud Radio Access Networks

This work aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN), where data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. We formulate a new mixed-integer nonlinear problem in which the ratio of network throughput and total power consumption is maximized. This challenging problem formulation includes practical constraints on routing, predefined minimum data rates, fronthaul capacity and maximum RRH transmit power. By employing the successive convex quadratic programming framework, an iterative algorithm is proposed with guaranteed convergence to a Fritz John solution of the formulated problem. Significantly, each iteration of the proposed algorithm solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization design markedly improves the C-RAN's energy efficiency compared to benchmark schemes.This work is supported in part by an ECR-HDR scholarship from The University of Newcastle, in part by the Australian Research Council Discovery Project grants DP170100939 and DP160101537, in part by Vietnam National Foundation for Science and Technology Development under grant number 101.02-2016.11 and in part by a startup fund from San Diego State University

Conditional Support Alignment for Domain Adaptation with Label Shift

Unsupervised domain adaptation (UDA) refers to a domain adaptation framework in which a learning model is trained based on the labeled samples on the source domain and unlabelled ones in the target domain. The dominant existing methods in the field that rely on the classical covariate shift assumption to learn domain-invariant feature representation have yielded suboptimal performance under the label distribution shift between source and target domains. In this paper, we propose a novel conditional adversarial support alignment (CASA) whose aim is to minimize the conditional symmetric support divergence between the source's and target domain's feature representation distributions, aiming at a more helpful representation for the classification task. We also introduce a novel theoretical target risk bound, which justifies the merits of aligning the supports of conditional feature distributions compared to the existing marginal support alignment approach in the UDA settings. We then provide a complete training process for learning in which the objective optimization functions are precisely based on the proposed target risk bound. Our empirical results demonstrate that CASA outperforms other state-of-the-art methods on different UDA benchmark tasks under label shift conditions