Perfect zero knowledge for quantum multiprover interactive proofs

In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP$^*$, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZKMIP$^*$, which is the set of languages decidable by MIP$^*$ protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP$^* =$ PZKMIP$^*$. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP $=$ PZKMIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP$^*$ protocol can be efficiently transformed into an equivalent zero knowledge MIP$^*$ protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results by Slofstra (Forum of Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we obtain the corollary that all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP$^*$ protocols with vanishing promise gap

Hybrid Vigor: An Analysis of Land Tenure Arrangements in Addressing Land Security for Urban Community Gardens

Urban community gardens (UCGs) are receiving greater attention as an ever-growing body of research documents the economic, environmental and social benefits that community gardens are bringing to urban neighborhoods. However, land insecurity remains one of the critical barriers to the future success of UCGs; loss of land to both private and public entities has frustrated many efforts and has stimulated investigations into alternative strategies for increasing land security. This thesis explores the diversity of land tenure solutions -- with a particular respect to land trusts -- that have been implemented to address this issue. Land security is objectively defined by legal property rights and subjectively shaped through the notion of 'ownership in use' and by encouraging the formation of mobilized communities. We analyze the variety of organizational modes -- differentiated by tenurial arrangement, organizational capacity, political leverage and organizational mission/internal governance relations -- and the resulting impacts on the type and level of land security. Data was collected through transcribed interviews with practitioners from sixteen land trust organizations engaged in urban community gardening and summarized along emergent themes regarding organizational modes and resulting land security implications. We find that major tradeoffs exist between maximizing land tenure security and operating a larger number of UCG sites under less secure tenure arrangements. It concludes that underutilized opportunities exist to increase UCG land security through an increased public sector role and broader cross-sectoral partnerships

Energy Management for a User Interactive Smart Community: A Stackelberg Game Approach

This paper studies a three party energy management problem in a user interactive smart community that consists of a large number of residential units (RUs) with distributed energy resources (DERs), a shared facility controller (SFC) and the main grid. A Stackelberg game is formulated to benefit both the SFC and RUs, in terms of incurred cost and achieved utility respectively, from their energy trading with each other and the grid. The properties of the game are studied and it is shown that there exists a unique Stackelberg equilibrium (SE). A novel algorithm is proposed that can be implemented in a distributed fashion by both RUs and the SFC to reach the SE. The convergence of the algorithm is also proven, and shown to always reach the SE. Numerical examples are used to assess the properties and effectiveness of the proposed scheme.Comment: 6 pages, 4 figure

Feasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid

This paper investigates the feasibility of using a discriminate pricing scheme to offset the inconvenience that is experienced by an energy user (EU) in trading its energy with an energy controller in smart grid. The main objective is to encourage EUs with small distributed energy resources (DERs), or with high sensitivity to their inconvenience, to take part in the energy trading via providing incentive to them with relatively higher payment at the same time as reducing the total cost to the energy controller. The proposed scheme is modeled through a two-stage Stackelberg game that describes the energy trading between a shared facility authority (SFA) and EUs in a smart community. A suitable cost function is proposed for the SFA to leverage the generation of discriminate pricing according to the inconvenience experienced by each EU. It is shown that the game has a unique sub-game perfect equilibrium (SPE), under the certain condition at which the SFA's total cost is minimized, and that each EU receives its best utility according to its associated inconvenience for the given price. A backward induction technique is used to derive a closed form expression for the price function at SPE, and thus the dependency of price on an EU's different decision parameters is explained for the studied system. Numerical examples are provided to show the beneficial properties of the proposed scheme.Comment: 7 pages, 4 figures, 3 tables, conference pape

Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar