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Proof-of-stake (PoS) is a promising approach for designing efficient blockchains, where block proposers are randomly chosen with probability proportional to their stake. A primary concern with PoS systems is the "rich getting richer" phenomenon, whereby wealthier nodes are more likely to get elected, and hence reap the block reward, making them even wealthier. In this paper, we introduce the notion of equitability, which quantifies how much a proposer can amplify her stake compared to her initial investment. Even with everyone following protocol (i.e., honest behavior), we show that existing methods of allocating block rewards lead to poor equitability, as does initializing systems with small stake pools and/or large rewards relative to the stake pool. We identify a \emph{geometric} reward function, which we prove is maximally equitable over all choices of reward functions under honest behavior and bound the deviation for strategic actions; the proofs involve the study of optimization problems and stochastic dominances of Polya urn processes, and are of independent mathematical interest. These results allow us to provide a systematic framework to choose the parameters of a practical incentive system for PoS cryptocurrencies., Comment: 26 pages

We report new observations of Abell 2256 with the Karl G. Jansky Very Large Array (VLA) at frequencies between 1 and 8 GHz. These observations take advantage of the 2:1 bandwidths available for a single observation to study the spectral index, polarization and Rotation Measure as well as using the associated higher sensitivity to image total intensity features down to ~0.5" resolution. We find the Large Relic, which dominates the cluster, is made up of a complex of filaments which show correlated distributions in intensity, spectral index, and fractional polarization. The Rotation Measure varies across the face of the Large Relic but is not well correlated with the other properties of the source. The shape of individual filaments suggests that the Large Relic is at least 25 kpc thick. We detect a low surface brightness arc connecting the Large Relic to the Halo and other radio structures suggesting a physical connection between these features. The center of the F-complex is dominated by a very steep-spectrum, polarized, ring-like structure, F2, without an obvious optical identification, but the entire F-complex has interesting morphological similarities to the radio structure of NGC1265. Source C, the Long Tail, is unresolved in width near the galaxy core and is

In this paper, we consider a class of stochastic optimal control problems with risk constraints that are expressed as bounded probabilities of failure for particular initial states. We present here a martingale approach that diffuses a risk constraint into a martingale to construct time-consistent control policies. The martingale stands for the level of risk tolerance over time. By augmenting the system dynamics with the controlled martingale, the original risk-constrained problem is transformed into a stochastic target problem. We extend the incremental Markov Decision Process (iMDP) algorithm to approximate arbitrarily well an optimal feedback policy of the original problem by sampling in the augmented state space and computing proper boundary conditions for the reformulated problem. We show that the algorithm is both probabilistically sound and asymptotically optimal. The performance of the proposed algorithm is demonstrated on motion planning and control problems subject to bounded probability of collision in uncertain cluttered environments.