Generalized prisoner's dilemma

Prisoner's dilemma has been heavily studied. In classical model, each player chooses to either "Cooperate" or "Defect". In this paper, we generalize the prisoner's dilemma with a new alternative which is neither defect or cooperation. The classical model is the special case under the condition that the third state is not taken into consideration.Comment: 7 pages, 2 figure

Revisiting Optimal Power Control: its Dual Effect on SNR and Contention

In this paper we study a transmission power tune problem with densely deployed 802.11 Wireless Local Area Networks (WLANs). While previous papers emphasize on tuning transmission power with either PHY or MAC layer separately, optimally setting each Access Point's (AP's) transmission power of a densely deployed 802.11 network considering its dual effects on both layers remains an open problem. In this work, we design a measure by evaluating impacts of transmission power on network performance on both PHY and MAC layers. We show that such an optimization problem is intractable and then we investigate and develop an analytical framework to allow simple yet efficient solutions. Through simulations and numerical results, we observe clear benefits of the dual-effect model compared to solutions optimizing solely on a single layer; therefore, we conclude that tuning transmission power from a dual layer (PHY-MAC) point of view is essential and necessary for dense WLANs. We further form a game theoretical framework and investigate above power-tune problem in a strategic network. We show that with decentralized and strategic users, the Nash Equilibrium (N.E.) of the corresponding game is in-efficient and thereafter we propose a punishment based mechanism to enforce users to adopt the social optimal strategy profile under both perfect and imperfect sensing environments

Recovery of an embedded obstacle and the surrounding medium for Maxwell's system

In this paper, we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric far-field data. The complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting (PEC) obstacle. The far-field data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation, but frequencies from an open interval. It is shown that the embedded obstacle can be uniquely recovered by the aforementioned far-field data, independent of the surrounding medium. Furthermore, if the surrounding medium is piecewise homogeneous, then the medium can be recovered as well. Those unique recovery results are new to the literature. Our argument is based on low-frequency expansions of the electromagnetic fields and certain harmonic analysis techniques.Comment: 15 page

Hardy Spaces (1<p<∞1<p<\infty) over Lipschitz Domains

Let $\Gamma$ be a Lipschitz curve on the complex plane $\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of holomorphic functions $F$ satisfying $\sup_{\tau>0}(\int_{\Gamma} |F(\zeta+\mathrm{i}\tau)|^p |\,\mathrm{d}\zeta|)^{\frac1p}< \infty$. We mainly focus on the case of $1<p<\infty$ in this paper, and prove that if $F(w)\in H^p(\Omega_+)$, then $F(w)$ has non-tangential boundary limit $F(\zeta)$ a.e. on $\Gamma$, and $F(w)$ is the Cauchy integral of $F(\zeta)$. We denote the conformal mapping from $\mathbb{C}_+$ onto $\Omega_+$ as $\Phi$, and then prove that, $H^p(\Omega_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on upper half plane, under the mapping $T\colon F\to F(\Phi(z))\cdot (\Phi'(z))^\frac{1}{p}$, where $F\in H^p(\Omega_+)$.Comment: 25 page

Quantum games of opinion formation based on the Marinatto-Weber quantum game scheme

Quantization becomes a new way to study classical game theory since quantum strategies and quantum games have been proposed. In previous studies, many typical game models, such as prisoner's dilemma, battle of the sexes, Hawk-Dove game, have been investigated by using quantization approaches. In this paper, several game models of opinion formations have been quantized based on the Marinatto-Weber quantum game scheme, a frequently used scheme to convert classical games to quantum versions. Our results show that the quantization can change fascinatingly the properties of some classical opinion formation game models so as to generate win-win outcomes.Comment: 19 page

A quantum extension to inspection game

Quantum game theory is a new interdisciplinary field between game theory and physical research. In this paper, we extend the classical inspection game into a quantum game version by quantizing the strategy space and importing entanglement between players. Our result shows that the quantum inspection game has various Nash equilibrium depending on the initial quantum state of the game. It is also shown that quantization can respectively help each player to increase his own payoff, yet fails to bring Pareto improvement for the collective payoff in the quantum inspection game.Comment: 6 page

Learning-based Prediction, Rendering and Association Optimization for MEC-enabled Wireless Virtual Reality (VR) Network

Wireless-connected Virtual Reality (VR) provides immersive experience for VR users from any-where at anytime. However, providing wireless VR users with seamless connectivity and real-time VR video with high quality is challenging due to its requirements in high Quality of Experience (QoE) and low VR interaction latency under limited computation capability of VR device. To address these issues,we propose a MEC-enabled wireless VR network, where the field of view (FoV) of each VR user can be real-time predicted using Recurrent Neural Network (RNN), and the rendering of VR content is moved from VR device to MEC server with rendering model migration capability. Taking into account the geographical and FoV request correlation, we propose centralized and distributed decoupled Deep Reinforcement Learning (DRL) strategies to maximize the long-term QoE of VR users under the VR interaction latency constraint. Simulation results show that our proposed MEC rendering schemes and DRL algorithms substantially improve the long-term QoE of VR users and reduce the VR interaction latency compared to rendering at VR device

Notes of Boundedness on Cauchy Integrals on Lipschitz Curves (p=2p=2)

We provide the details of the first proof in~\cite{CJS89}, which proved that Cauchy transform of $L^2$~functions on Lipschitz curves is bounded. We then prove that every $L^2$~function on Lipschitz curves is the sum of non-tangential boundary limit of functions in $H^2(\Omega_\pm)$, the Hardy spaces on domains over and under the Lipschitz curve. We also obtain a more accurate boundary of Cauchy transform under the condition that the Lipschitz curve is the real axis.Comment: 22 page

First exit and Dirichlet problem for the nonisotropic tempered α\alpha-stable processes

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $\alpha$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{\tau_D}\right|$ and the first exit time $\tau_D$ are firstly obtained. It is found that the probability density function of $\left|X_{\tau_D}\right|$ or $\tau_D$ exponentially decays with the increase of $\left|X_{\tau_D}\right|$ or $\tau_D$, and $\mathrm{E}\left[\tau_D\right]\sim \left|\mathrm{E}\left[X_{\tau_D}\right]\right|$,\ $\mathrm{E}\left[\tau_D\right]\sim\mathrm{E}\left[\left|X_{\tau_D}-\mathrm{E}\left[X_{\tau_D}\right]\right|^2\right]$. Since $\mathrm{\Delta}^{\alpha/2,\lambda}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrm{\Delta}^{\alpha/2,\lambda}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.Comment: 23 pages, 5 figure

Hardy Spaces over Half-strip Domains

We define Hardy spaces $H^p(\Omega_\pm)$ on half-strip domain~$\Omega_+$ and $\Omega_-= \mathbb{C}\setminus\overline{\Omega_+}$, where $0<p<\infty$, and prove that functions in $H^p(\Omega_\pm)$ has non-tangential boundary limit a.e. on $\Gamma$, the common boundary of $\Omega_\pm$. We then prove that Cauchy integral of functions in $L^p(\Gamma)$ are in $H^p(\Omega_\pm)$, where $1<p<\infty$, that is, Cauchy transform is bounded. Besides, if $1\leqslant p<\infty$, then $H^p(\Omega_\pm)$ functions are the Cauchy integral of their non-tangential boundary limits. We also establish an isomorphism between $H^p(\Omega_\pm)$ and $H^p(\mathbb{C}_\pm)$, the classical Hardy spaces over upper and lower half complex planes.Comment: 38 page