Lehmer points and visible points on affine varieties over finite fields

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.Comment: 16 pages, accepted for publication in Math. Proc. Cambridge Philos. Soc. Theorem 2 of this paper is improved from the published versio

Optimal Energy Allocation for Wireless Communications with Energy Harvesting Constraints

We consider the use of energy harvesters, in place of conventional batteries with fixed energy storage, for point-to-point wireless communications. In addition to the challenge of transmitting in a channel with time selective fading, energy harvesters provide a perpetual but unreliable energy source. In this paper, we consider the problem of energy allocation over a finite horizon, taking into account channel conditions and energy sources that are time varying, so as to maximize the throughput. Two types of side information (SI) on the channel conditions and harvested energy are assumed to be available: causal SI (of the past and present slots) or full SI (of the past, present and future slots). We obtain structural results for the optimal energy allocation, via the use of dynamic programming and convex optimization techniques. In particular, if unlimited energy can be stored in the battery with harvested energy and the full SI is available, we prove the optimality of a water-filling energy allocation solution where the so-called water levels follow a staircase function.Comment: 27 pages, 6 figures, accepted for publications at IEEE Transactions on Signal Processin

The Collapsing Rate of the K\"ahler-Ricci Flow with Regular Infinite Time Singularity

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with possibly singular fibers will also be discussed in local sense.Comment: 18 pages; final version, to appear in J. Reine Angew. Mat

Optimal Quantization in Energy-Constrained Sensor Networks under Imperfect Transmission

This paper addresses the optimization of quantization at local sensors under strict energy constraint and imperfect transmission to improve the reconstruction performance at the fusion center in the wireless sensor networks (WSNs). We present optimized quantization scheme including the optimal quantization bit rate and the optimal transmission power allocation among quantization bits for BPSK signal and binary orthogonal signal with envelope detection, respectively. The optimization of the quantization is formulated as a convex problem and the optimal solution is derived analytically in both cases. Simulation results demonstrate the effectiveness of our proposed quantization schemes

Wireless Information and Power Transfer: Architecture Design and Rate-Energy Tradeoff

Simultaneous information and power transfer over the wireless channels potentially offers great convenience to mobile users. Yet practical receiver designs impose technical constraints on its hardware realization, as practical circuits for harvesting energy from radio signals are not yet able to decode the carried information directly. To make theoretical progress, we propose a general receiver operation, namely, dynamic power splitting (DPS), which splits the received signal with adjustable power ratio for energy harvesting and information decoding, separately. Three special cases of DPS, namely, time switching (TS), static power splitting (SPS) and on-off power splitting (OPS) are investigated. The TS and SPS schemes can be treated as special cases of OPS. Moreover, we propose two types of practical receiver architectures, namely, separated versus integrated information and energy receivers. The integrated receiver integrates the front-end components of the separated receiver, thus achieving a smaller form factor. The rate-energy tradeoff for the two architectures are characterized by a so-called rate-energy (R-E) region. The optimal transmission strategy is derived to achieve different rate-energy tradeoffs. With receiver circuit power consumption taken into account, it is shown that the OPS scheme is optimal for both receivers. For the ideal case when the receiver circuit does not consume power, the SPS scheme is optimal for both receivers. In addition, we study the performance for the two types of receivers under a realistic system setup that employs practical modulation. Our results provide useful insights to the optimal practical receiver design for simultaneous wireless information and power transfer (SWIPT).Comment: to appear in IEEE Transactions on Communication

Does Format of Pricing Contract Matter?

The use of linear wholesale price contract has long been recognized as a threat to achieving channel effciency. Many formats of nonlinear pricing contract have been proposed to achieve vertical channel coordination. Examples include two-part tariff and quantity discount. A two-part tariff charges the downstream party a fixed fee for participation and a uniform unit price. A quantity discount contract does not include a fixed fee and charges a lower unit price for each additional unit. Extant economic theories predict these contracts, when chosen optimally, to be revenue and division equivalent in that they all restore full channel effciency and give the same surplus to the upstream party assuming constant relative bargaining power. We conduct a laboratory experiment to test the empirical equivalence of the two pricing formats. Surprisingly, both pricing formats fail to coordinate the channel even in a well-controlled market environment with subjects motivated by significant monetary incentives. The observed channl effciencies were significantly lower than 100%. In fact, they are statistically no better than that of the linear wholesale price contract. Revenue equivalence fails because the quantity discount scheme achieves a higher channel effciency than the two-part tariff. Also, division equivalence does not hold because the quantity discount scheme accords a higher surplus to the upstream party than the two-part tariff. To account for the observed empirical regularities, we allow the downstream party to have a reference-dependent utility in which the upfront fixed fee is framed as loss andn the subsequent contribution margin as gain. The proposed model nests the standard economic model as a special case with a loss aversion coeffcient of 1.0. The estimated loss aversion coeffcient is 1.6, thereby rejecting the standard model. We rule out other plausible explanations such as parties having fairness concerns and non-linear risk attitudes.Pricing Format, Two-Part Tariff, Quantity Discount, Channel Efficiency, Double Marginalization, Reference-Dependent Utility, Experimental Economics, Behavioral Economics

Recent Advances in Joint Wireless Energy and Information Transfer

In this paper, we provide an overview of the recent advances in microwave-enabled wireless energy transfer (WET) technologies and their applications in wireless communications. Specifically, we divide our discussions into three parts. First, we introduce the state-of-the-art WET technologies and the signal processing techniques to maximize the energy transfer efficiency. Then, we discuss an interesting paradigm named simultaneous wireless information and power transfer (SWIPT), where energy and information are jointly transmitted using the same radio waveform. At last, we review the recent progress in wireless powered communication networks (WPCN), where wireless devices communicate using the power harvested by means of WET. Extensions and future directions are also discussed in each of these areas.Comment: Conference submission accepted by ITW 201