Expected Fair Allocation in Farsighted Network Formation

I consider cooperation situations where players have network relations. Networks evolve according to a stationary transition probability matrix and at each moment in time players receive payoffs from a stationary allocation rule. Players discount the future by a common factor. The pair formed by an allocation rule and a transition probability matrix is called expected fair if for every link in the network both participants gain, marginally, and in discounted, expected terms, the same from it; and it is called a pairwise network formation procedure if the probability that a link is created (or eliminated) is positive if the discounted, expected gains to its two participants are positive too. The main result is the existence, for the discount factor small enough, of an expected fair and pairwise network formation procedure where the allocation rule is component balanced, meaning it distributes the total value of any maximal connected subnetwork among its participants. This existence result holds for all discount factors when the pairwise network formation procedure is restricted. I finally provide some comparison with previous models of farsighted network formation.

Forward-looking Pairwise Stability in Networks with Externalities

We consider cooperation situations where players have network relations. Networks evolve according to a stationary transition probability matrix and at each moment in time players receive payoffs from a stationary allocation rule. Players discount the future by a common factor. The pair formed by an allocation rule and a transition probability matrix is called a forward-looking network formation scheme if, first, the probability that a link is created is positive if the discounted, expected gains to its two participants are positive, and if, second, the probability that a link is eliminated is positive if the discounted, expected gains to at least one of its two participants are positive. The main result is the existence, for all discount factors and all value functions, of a forward-looking network formation scheme. Furthermore, we can always nd a forward-looking network formation scheme such that (i) the allocation rule is component balanced and (ii) the transition probabilities increase in the di erence in payo s for the corresponding players responsible for the transition. We use this dynamic solution concept to explore the tension between e ciency and stability.

Radiotherapy plus nimotuzumab or placebo in the treatment of high grade glioma patients: results from a randomized, double blind trial

Background The prognosis of patients bearing high grade glioma remains dismal. Epidermal Growth Factor Receptor (EGFR) is well validated as a primary contributor of glioma initiation and progression. Nimotuzumab is a humanized monoclonal antibody that recognizes the EGFR extracellular domain and reaches Central Nervous System tumors, in nonclinical and clinical setting. While it has similar activity when compared to other anti-EGFR antibodies, it does not induce skin toxicity or hypomagnesemia. Methods A randomized, double blind, multicentric clinical trial was conducted in high grade glioma patients (41 anaplastic astrocytoma and 29 glioblastoma multiforme) that received radiotherapy plus nimotuzumab or placebo. Treatment and placebo groups were well-balanced for the most important prognostic variables. Patients received 6 weekly doses of 200 mg nimotuzumab or placebo together with irradiation as induction therapy. Maintenance treatment was given for 1 year with subsequent doses administered every 3 weeks. The objectives of this study were to assess the comparative overall survival, progression free survival, response rate, immunogenicity and safety. Results The median cumulative dose was 3200 mg of nimotuzumab given over a median number of 16 doses. The combination of nimotuzumab and RT was well-tolerated. The most prevalent related adverse reactions included nausea, fever, tremors, anorexia and hepatic test alteration. No anti-idiotypic response was detected, confirming the antibody low immunogenicity. The mean and median survival time for subjects treated with nimotuzumab was 31.06 and 17.76 vs. 21.07 and 12.63 months for the control group. Conclusions In this randomized trial, nimotuzumab showed an excellent safety profile and significant survival benefit in combination with irradiation.

The cross layer RMPA handover: a reliable mobility pattern aware handover strategy for broadband wireless communication in a high-speed railway domain

Enhancing the handover process in broadband wireless communication deployment has traditionally motivated many research initiatives. In a high-speed railway domain, the challenge is even greater. Owing to the long distances covered, the mobile node gets involved in a compulsory sequence of handover processes. Consequently, poor performance during the execution of these handover processes significantly degrades the global end-to-end performance. This article proposes a new handover strategy for the railway domain: the RMPA handover, a Reliable Mobility Pattern Aware IEEE 802.16 handover strategy "customized" for a high-speed mobility scenario. The stringent high mobility feature is balanced with three other positive features in a high-speed context: mobility pattern awareness, different sources for location discovery techniques, and a previously known traffic data profile. To the best of the authors' knowledge, there is no IEEE 802.16 handover scheme that simultaneously covers the optimization of the handover process itself and the efficient timing of the handover process. Our strategy covers both areas of research while providing a cost-effective and standards-based solution. To schedule the handover process efficiently, the RMPA strategy makes use of a context aware handover policy; that is, a handover policy based on the mobile node mobility pattern, the time required to perform the handover, the neighboring network conditions, the data traffic profile, the received power signal, and current location and speed information of the train. Our proposal merges all these variables in a cross layer interaction in the handover policy engine. It also enhances the handover process itself by establishing the values for the set of handover configuration parameters and mechanisms of the handover process. RMPA is a cost-effective strategy because compatibility with standards-based equipment is guaranteed. The major contributions of the RMPA handover are in areas that have been left open to the handover designer's discretion. Our simulation analysis validates the RMPA handover decision rules and design choices. Our results supporting a high-demand video application in the uplink stream show a significant improvement in the end-to-end quality of service parameters, including end-to-end delay (22%) and jitter (80%), when compared with a policy based on signal-to-noise-ratio information.

San Francisco Bay regional sediment management strategy development

The San Francisco Bay Conservation and Development Commission (BCDC), in continued partnership with the San Francisco Bay Long Term Management Strategies (LTMS) Agencies, is undertaking the development of a Regional Sediment Management Plan for the San Francisco Bay estuary and its watershed (estuary). Regional sediment management (RSM) is the integrated management of littoral, estuarine, and riverine sediments to achieve balanced and sustainable solutions to sediment related needs. Regional sediment management recognizes sediment as a resource. Sediment processes are important components of coastal and riverine systems that are integral to environmental and economic vitality. It relies on the context of the sediment system and forecasting the long-range effects of management actions when making local project decisions. In the San Francisco Bay estuary, the sediment system includes the Sacramento and San Joaquin delta, the bay, its local tributaries and the near shore coastal littoral cell. Sediment flows from the top of the watershed, much like water, to the coast, passing through rivers, marshes, and embayments on its way to the ocean. Like water, sediment is vital to these habitats and their inhabitants, providing nutrients and the building material for the habitat itself. When sediment erodes excessively or is impounded behind structures, the sediment system becomes imbalanced, and rivers become clogged or conversely, shorelines, wetlands and subtidal habitats erode. The sediment system continues to change in response both to natural processes and human activities such as climate change and shoreline development. Human activities that influence the sediment system include flood protection programs, watershed management, navigational dredging, aggregate mining, shoreline development, terrestrial, riverine, wetland, and subtidal habitat restoration, and beach nourishment. As observed by recent scientific analysis, the San Francisco Bay estuary system is changing from one that was sediment rich to one that is erosional. Such changes, in conjunction with increasing sea level rise due to climate change, require that the estuary sediment and sediment transport system be managed as a single unit. To better manage the system, its components, and human uses of the system, additional research and knowledge of the system is needed. Fortunately, new sediment science and modeling tools provide opportunities for a vastly improved understanding of the sediment system, predictive capabilities and analysis of potential individual and cumulative impacts of projects. As science informs management decisions, human activities and management strategies may need to be modified to protect and provide for existing and future infrastructure and ecosystem needs. (PDF contains 3 pages)

The island of Kauai, Hawaii's progressive shoreline setback and coastal protection ordinance

Approximately two-thirds of coastal and Great Lakes states have some type of shoreline construction setback or construction control line requiring development to be a certain distance from the shoreline or other coastal feature (OCRM, 2008). Nineteen of 30 coastal states currently use erosion rates for new construction close to the shoreline. Seven states established setback distances based on expected years from the shoreline: the remainder specify a fixed setback distance (Heinz Report, 2000). Following public hearings by the County of Kauai Planning Commission and Kauai County Council, the ‘Shoreline Setback and Coastal Protection Ordinance’ was signed by the Mayor of Kauai on January 25, 2008. After a year of experience implementing this progressive, balanced shoreline setback ordinance several amendments were recently incorporated into the Ordinance (#887; Bill #2319 Draft 3). The Kauai Planning Department is presently drafting several more amendments to improve the effectiveness of the Ordinance. The intent of shoreline setbacks is to establish a buffer zone to protect shorefront development from loss due to coastal erosion - for a period of time; to provide protection from storm waves; to allow the natural dynamic cycles of erosion and accretion of beaches and dunes to occur; to maintain beach and dune habitat; and, to maintain lateral beach access and open space for the enjoyment of the natural shoreline environment. In addition, a primary goal of the Kauai setback ordinance is to avoid armoring or hardening of the shore which along eroding coasts has been documented to ultimately eliminate the fronting beach. (PDF contains 4 pages)

Characterizing distribution rules for cost sharing games

In noncooperative cost sharing games, individually strategic agents choose resources based on how the welfare (cost or revenue) generated at each resource (which depends on the set of agents that choose the resource) is distributed. The focus is on finding distribution rules that lead to stable allocations, which is formalized by the concept of Nash equilibrium, e.g., Shapley value (budget-balanced) and marginal contribution (not budget-balanced) rules.

Recent work that seeks to characterize the space of all such rules shows that the only budget-balanced distribution rules that guarantee equilibrium existence in all welfare sharing games are generalized weighted Shapley values (GWSVs), by exhibiting a specific 'worst-case' welfare function which requires that GWSV rules be used. Our work provides an exact characterization of the space of distribution rules (not necessarily budget-balanced) for any specific local welfare functions remains, for a general class of scalable and separable games with well-known applications, e.g., facility location, routing, network formation, and coverage games.

We show that all games conditioned on any fixed local welfare functions possess an equilibrium if and only if the distribution rules are equivalent to GWSV rules on some 'ground' welfare functions. Therefore, it is neither the existence of some worst-case welfare function, nor the restriction of budget-balance, which limits the design to GWSVs. Also, in order to guarantee equilibrium existence, it is necessary to work within the class of potential games, since GWSVs result in (weighted) potential games.

We also provide an alternative characterization—all games conditioned on any fixed local welfare functions possess an equilibrium if and only if the distribution rules are equivalent to generalized weighted marginal contribution (GWMC) rules on some 'ground' welfare functions. This result is due to a deeper fundamental connection between Shapley values and marginal contributions that our proofs expose—they are equivalent given a transformation connecting their ground welfare functions. (This connection leads to novel closed-form expressions for the GWSV potential function.) Since GWMCs are more tractable than GWSVs, a designer can tradeoff budget-balance with computational tractability in deciding which rule to implement.

Parents aspiration and youth perception to sustainable fisheries development in Kainji Lake basin

Artisanal fisheries development in Nigeria, like in any other developing country of the world is characterized by subsistence level of operation using dugout canoe and paddle. This implies that parents and children constituted the labour in use since they only struggle for the upkeep of the family. A total of 240 questionnaires were used to solicit information from the respondents. This total was divided into 120 each for both parents and the youths respectively. Simple descriptive statistics such as frequency distribution and percentages were used to analyze their responses. Parents have strong aspiration for their children to succeed them hence, 78.3% of the parents expressed their aspiration for succession by the younger ones while 83.3% of parents tend to support their aspiration with persuasive strategies such as allowing their children to partake in fishing activities at will. On the part of the youth over 70% of them perceived fishing as a viable business where they can succeed their parents provided government can come to their aid in form of active involvement in the development of rural fisheries. It is therefore believed that fisheries development could achieve rapid improvement if the aspiration of the parents is balanced with perception of the youth and the government meets their expectation

Magnetohydrodynamic turbulence

We simulate incompressible, MHD turbulence using a pseudo-spectral code. Our major conclusions are as follows.

1) MHD turbulence is most conveniently described in terms of counter propagating shear Alfvén and slow waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves.

2) MHD turbulence is anisotropic with energy cascading more rapidly along k_⊥ than along k_∥, where k_⊥ and k_∥ refer to wavevector components perpendicular and parallel to the local magnetic field. Anisotropy increases with increasing k_⊥ such that excited modes are confined inside a cone bounded by k_∥ ∝ k^γ_⊥ where γ less than 1. The opening angle of the cone, θ(k_⊥) ∝ k^-(1-γ)_⊥, defines the scale dependent anisotropy.

3) MHD turbulence is generically strong in the sense that the waves which comprise it suffer order unity distortions on timescales comparable to their periods. Nevertheless, turbulent fluctuations are small deep inside the inertial range. Their energy density is less than that of the background field by a factor θ² (k_⊥)≪1.

4) MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating waves. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The amplitude of the former exceeds that of the latter by 1/θ(k_⊥) which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics.

5) Passive scalars mixed by MHD turbulence adopt the same power spectrum as the velocity and magnetic field perturbations.

6) Decaying MHD turbulence is unstable to an increase of the imbalance between the flux of waves propagating in opposite directions along the magnetic field. Forced MHD turbulence displays order unity fluctuations with respect to the balanced state if excited at low k by δ(t) correlated forcing. It appears to be statistically stable to the unlimited growth of imbalance.

7) Gradients of the dynamic variables are focused into sheets aligned with the magnetic field whose thickness is comparable to the dissipation scale. Sheets formed by oppositely directed waves are uncorrelated. We suspect that these are vortex sheets which the mean magnetic field prevents from rolling up.

8) Items (1)-(5) lend support to the model of strong MHD turbulence put forth by Goldreich and Sridhar (1995, 1997). Results from our simulations are also consistent with the GS prediction γ = 2/3. The sole not able discrepancy is that the 1D power law spectra, E(k_⊥) ∝ k^-∝_⊥, determined from our simulations exhibit ∝ ≈ 3/2, whereas the GS model predicts ∝ = 5/3.

Label-free detection of single biological molecules using microtoroid optical resonators

Being able to detect a single molecule without the use of labels has been a long standing goal of bioengineers and physicists. This would simplify applications ranging from single molecular binding studies to those involving public health and security, improved drug screening, medical diagnostics, and genome sequencing. One promising technique that has the potential to detect single molecules is the microtoroid optical resonator. The main obstacle to detecting single molecules, however, is decreasing the noise level of the measurements such that a single molecule can be distinguished from background. We have used laser frequency locking in combination with balanced detection and data processing techniques to reduce the noise level of these devices and report the detection of a wide range of nanoscale objects ranging from nanoparticles with radii from 100 to 2.5 nm, to exosomes, ribosomes, and single protein molecules (mouse immunoglobulin G and human interleukin-2). We further extend the exosome results towards creating a non-invasive tumor biopsy assay. Our results, covering several orders of magnitude of particle radius (100 nm to 2 nm), agree with the `reactive' model prediction for the frequency shift of the resonator upon particle binding. In addition, we demonstrate that molecular weight may be estimated from the frequency shift through a simple formula, thus providing a basis for an ``optical mass spectrometer'' in solution. We anticipate that our results will enable many applications, including more sensitive medical diagnostics and fundamental studies of single receptor-ligand and protein-protein interactions in real time. The thesis summarizes what we have achieved thus far and shows that the goal of detecting a single molecule without the use of labels can now be realized.

Regiões metropolitanas brasileiras e autonomia municipal: eficácia da norma constitucional brasileira e de experiências estrangeiras na gestão dos interesses regionais

O atendimento às demandas de determinada parcela da população que vive nas chamadas Regiões Metropolitanas no Brasil tem apresentado acentuada dificuldade em obter resultados satisfatórios, na medida em que estes espaços territoriais estejam situados em diferentes jurisdições político-territoriais. Tais dificuldades têm origem, sobretudo, na necessidade da composição de arranjos governamentais que possam atuar de forma conjunta e coordenada, abrangendo Estados e Municípios envolvidos nesta dinâmica metropolitana, e que abrange aspectos fiscais, sociais, ambientais e jurídicos. O presente trabalho analisa este último aspecto, sobretudo, em relação à questão das competências constitucionais dos entes envolvidos e o papel a ser desempenhado por cada um na regulação do solo urbano, um dos aspectos mais relevantes em relação ao tema metropolitano. Se a dependência de um eventual acordo entre os entes federativos tem se mostrado raro na história federativa brasileira, tal fato não pode constituir-se em um fator impeditivo do alcance dos direitos fundamentais estabelecidos pela Constituição Federal, principalmente levando-se em consideração que uma regulação adequada do solo urbano em uma perspectiva regional (metropolitana) é uma meio fundamental para o alcance de vários direitos, como moradia, meio ambiente equilibrado. Identificando o Estado-Membro como figura principal deste mister, por meio de uma interpretação sistemática e teleológica da Constituição, e reconhecendo o cenário de constitucionalização do direito administrativo atual bem como da chamada crise da lei, verifica-se que este ente federativo pode e deve assumir plenamente suas competências, elaborando um estudo técnico de planejamento regional, não necessariamente aprovado por lei formal, e vinculante para os Municípios.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.

Comparative drift studies on streams in lower Austria (Flysch, gneiss and chalk formations) [Translation from: Schweizerische Zeitschrift fur Hydrobiologie 30 138-185, 1968.]

Seasonal changes and flooding have an extraordinarily great influence on the drift of organisms. The free water space plays the main part in the provision of food for some fish (Salmo trutta - trout): drift and content of the stomach are balanced here (Simuliidae): whereas others (Thymallus vulgaris) only selectively chose certain animals living at the bottom (molluscs). The total drift, drift of organisms and drift of organic material and minerals, plays a main role in the rate of production in streams. Besides the biology of the organisms living on the river bed, also the geological and hydrographical situation of the area plays a very important role for the composition of the drift. During the years 1964-1966 three streams in the characteristical geological formations flysch, gneiss and chalk of lower Austria were studied in regard to their drift. The Tulln (above St. Christopen), the Krems (above Senftenberg) and the Schwarza (above Hirschwang) seemed to be ideal for this comparative study because they are easy to reach. After summarising the hydrography and chemistry of examined rivers, the author examines the relationship between water level and total drift and the stratification of the total drift before analysing the drift of living organisms. Also considered are seasonal changes of drift of organisms and drift of exuviae.

Distributed Control and Optimization for Communication and Power Systems

We are at the cusp of a historic transformation of both communication system and electricity system. This creates challenges as well as opportunities for the study of networked systems. Problems of these systems typically involve a huge number of end points that require intelligent coordination in a distributed manner. In this thesis, we develop models, theories, and scalable distributed optimization and control algorithms to overcome these challenges.

This thesis focuses on two specific areas: multi-path TCP (Transmission Control Protocol) and electricity distribution system operation and control. Multi-path TCP (MP-TCP) is a TCP extension that allows a single data stream to be split across multiple paths. MP-TCP has the potential to greatly improve reliability as well as efficiency of communication devices. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new algorithm Balia (balanced linked adaptation) which generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We have implemented Balia in the Linux kernel. We use our prototype to compare the new proposed algorithm Balia with existing MP-TCP algorithms.

Our second focus is on designing computationally efficient algorithms for electricity distribution system operation and control. First, we develop efficient algorithms for feeder reconfiguration in distribution networks. The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. We propose a heuristic algorithm that is based on the recently developed convex relaxation of the optimal power flow problem. The algorithm is efficient and can successfully computes an optimal configuration on all networks that we have tested. Moreover we prove that the algorithm solves the feeder reconfiguration problem optimally under certain conditions. We also propose a more efficient algorithm and it incurs a loss in optimality of less than 3% on the test networks.

Second, we develop efficient distributed algorithms that solve the optimal power flow (OPF) problem on distribution networks. The OPF problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. Traditionally OPF is solved in a centralized manner. With increasing penetration of volatile renewable energy resources in distribution systems, we need faster and distributed solutions for real-time feedback control. This is difficult because power flow equations are nonlinear and kirchhoff's law is global. We propose solutions for both balanced and unbalanced radial distribution networks. They exploit recent results that suggest solving for a globally optimal solution of OPF over a radial network through a second-order cone program (SOCP) or semi-definite program (SDP) relaxation. Our distributed algorithms are based on the alternating direction method of multiplier (ADMM), but unlike standard ADMM-based distributed OPF algorithms that require solving optimization subproblems using iterative methods, the proposed solutions exploit the problem structure that greatly reduce the computation time. Specifically, for balanced networks, our decomposition allows us to derive closed form solutions for these subproblems and it speeds up the convergence by 1000x times in simulations. For unbalanced networks, the subproblems reduce to either closed form solutions or eigenvalue problems whose size remains constant as the network scales up and computation time is reduced by 100x compared with iterative methods.

A potential approach to claims problems

Hart and Mas Colell (1989) introduce the potential function for cooperative TU games. In this paper, we extend this approach to claims problems, also known as bankruptcy or rationing problems. We show that for appropriate subproblems, the random arrival rule, the rules in the TAL-family (which include the uniform gains rule, the uniform losses rule and the Talmud rule), the minimal overlap rule, and the proportional rule admit a potential. We also study the balanced contributions property for these rules. By means of a potential, we introduce a generalization of the random arrival rule and mixtures of the minimal overlap rule and the uniform losses rule.