Local models of Shimura varieties, I. Geometry and combinatorics

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the "Handbook of Moduli

Matching of analytical and numerical solutions for neutron stars of arbitrary rotation

We demonstrate the results of an attempt to match the two-soliton analytical solution with the numerically produced solutions of the Einstein field equations, that describe the spacetime exterior of rotating neutron stars, for arbitrary rotation. The matching procedure is performed by equating the first four multipole moments of the analytical solution to the multipole moments of the numerical one. We then argue that in order to check the effectiveness of the matching of the analytical with the numerical solution we should compare the metric components, the radius of the innermost stable circular orbit ($R_{ISCO}$), the rotation frequency $\Omega\equiv\frac{d\phi}{dt}$ and the epicyclic frequencies $\Omega_{\rho},\;\Omega_z$. Finally we present some results of the comparison.Comment: Contribution at the 13th Conference on Recent Developments in Gravity (NEB XIII), corrected typo in $M_4$ of eq. 5 of the published versio

On the Limited Communication Analysis and Design for Decentralized Estimation

This paper pertains to the analysis and design of decentralized estimation schemes that make use of limited communication. Briefly, these schemes equip the sensors with scalar states that iteratively merge the measurements and the state of other sensors to be used for state estimation. Contrarily to commonly used distributed estimation schemes, the only information being exchanged are scalars, there is only one common time-scale for communication and estimation, and the retrieval of the state of the system and sensors is achieved in finite-time. We extend previous work to a more general setup and provide necessary and sufficient conditions required for the communication between the sensors that enable the use of limited communication decentralized estimation~schemes. Additionally, we discuss the cases where the sensors are memoryless, and where the sensors might not have the capacity to discern the contributions of other sensors. Based on these conditions and the fact that communication channels incur a cost, we cast the problem of finding the minimum cost communication graph that enables limited communication decentralized estimation schemes as an integer programming problem.Comment: Updates on the paper in CDC 201

Decentralized Observability with Limited Communication between Sensors

In this paper, we study the problem of jointly retrieving the state of a dynamical system, as well as the state of the sensors deployed to estimate it. We assume that the sensors possess a simple computational unit that is capable of performing simple operations, such as retaining the current state and model of the system in its memory. We assume the system to be observable (given all the measurements of the sensors), and we ask whether each sub-collection of sensors can retrieve the state of the underlying physical system, as well as the state of the remaining sensors. To this end, we consider communication between neighboring sensors, whose adjacency is captured by a communication graph. We then propose a linear update strategy that encodes the sensor measurements as states in an augmented state space, with which we provide the solution to the problem of retrieving the system and sensor states. The present paper contains three main contributions. First, we provide necessary and sufficient conditions to ensure observability of the system and sensor states from any sensor. Second, we address the problem of adding communication between sensors when the necessary and sufficient conditions are not satisfied, and devise a strategy to this end. Third, we extend the former case to include different costs of communication between sensors. Finally, the concepts defined and the method proposed are used to assess the state of an example of approximate structural brain dynamics through linearized measurements.Comment: 15 pages, 5 figures, extended version of paper accepted at IEEE Conference on Decision and Control 201

Emission of Massive Scalar Fields by a Higher-Dimensional Rotating Black-Hole

We perform a comprehensive study of the emission of massive scalar fields by a higher-dimensional, simply rotating black hole both in the bulk and on the brane. We derive approximate, analytic results as well as exact numerical ones for the absorption probability, and demonstrate that the two sets agree very well in the low and intermediate-energy regime for scalar fields with mass m_\Phi < 1 TeV in the bulk and m_\Phi < 0.5 TeV on the brane. The numerical values of the absorption probability are then used to derive the Hawking radiation power emission spectra in terms of the number of extra dimensions, angular-momentum of the black hole and mass of the emitted field. We compute the total emissivities in the bulk and on the brane, and demonstrate that, although the brane channel remains the dominant one, the bulk-over-brane energy ratio is considerably increased (up to 33%) when the mass of the emitted field is taken into account.Comment: 28 pages, 18 figure

The Costs of Creating Environmental Markets: A Commodification Primer

Markets offer a potent tool for managing resources and values, even ones that have not traditionally been commodified. In the environmental context there is particular debate about market-based governance, in terms of both appropriateness and effectiveness. This Article offers a broadly applicable framework for considering the emergence, appropriateness, and design of market tools in environmental governance, and it demonstrates how the model is applicable well beyond that context. This framework offers a powerful diagnostic for programs to manage resources ranging from greenhouse gas emissions to Chesapeake Bay pollution, as well as from human organs to Uber regulation. As a foundation for this framework, the Article identifies and examines two sets of underappreciated costs associated with establishing and utilizing market mechanisms. It terms these costs “severance costs” and “adjustment failure costs.” Severance costs describe the costs associated with defining, enforcing, and transacting in marketable “goods.” For instance, to pluck an environmental good from its interconnected ecological and legal context and to attempt to define it as a severable, stand-alone commodity can be costly. Additionally, when such an environmental good is not necessarily associated with tangible, physical ownership or when it has not historically been commodified, further challenges arise in creating the complex institutions necessary for such markets to function. If severance costs are too high, property interests may never be defined or transactions may never occur. In addition to severance costs, “adjustment failure costs” inherent in the pricing system represent another critical set of considerations that impact the emergence and success of market mechanisms. In all markets, pricing results from an iterative trial-and-error process, and it takes time and misallocations for supply and demand to align (assuming they ever do). The adjustment failure costs associated with such pricing delays and corrections may be trivial in some markets, but they can be particularly high and material in the context of non-fungible or irreparable goods. Since environmental goods in particular may display such non-fungible or irreparable characteristics, consideration of adjustment failure costs is crucial for environmental market mechanisms because high adjustment failure costs may exceed the potential gains of the market system. Thus, the adjustment failure costs that arise from the iterative function of markets represent another key factor in determining the appropriateness and success of market tools. This Article posits that severance costs and adjustment failure costs represent the two most significant dimensions for assessing the appropriateness and design of market instruments, both in the environmental context and more broadly. If these costs are too high, either individually or in combination, they will exceed the potential gains of a market system. Based on these sets of costs, the Article constructs a model for evaluating market emergence and success, and with this framework, the Article makes two major contributions. First, it offers a concrete and pragmatic method for gauging the desirability of market tools for certain resources in the environmental context and beyond. For instance, the model can identify specific situations where a cap-and-trade approach will be less effective than a Pigouvian-tax, or where a licensing system will be superior to a laissez-faire one. Consideration of severance costs and adjustment failure costs offers a generalizable model for describing the feasibility of commodifying environmental goods, prescribing interventions to marginally improve market instruments in general, and evaluating governance approaches for a variety of contexts. Second, this Article contributes to the theoretical literature on commodification by offering a positive economic framework that can synthesize the leading scholarship and explain existing reservations regarding commodification. It provides a descriptive economic account that can help ground moral intuitions and objections about markets and commodification. As a result, it gives fresh insight into why existing laws and policies are as they are, and it bridges moral and economic arguments, providing a common point ofdeparture for future engagement in these debates

Cloud-based Quadratic Optimization with Partially Homomorphic Encryption

The development of large-scale distributed control systems has led to the outsourcing of costly computations to cloud-computing platforms, as well as to concerns about privacy of the collected sensitive data. This paper develops a cloud-based protocol for a quadratic optimization problem involving multiple parties, each holding information it seeks to maintain private. The protocol is based on the projected gradient ascent on the Lagrange dual problem and exploits partially homomorphic encryption and secure multi-party computation techniques. Using formal cryptographic definitions of indistinguishability, the protocol is shown to achieve computational privacy, i.e., there is no computationally efficient algorithm that any involved party can employ to obtain private information beyond what can be inferred from the party's inputs and outputs only. In order to reduce the communication complexity of the proposed protocol, we introduced a variant that achieves this objective at the expense of weaker privacy guarantees. We discuss in detail the computational and communication complexity properties of both algorithms theoretically and also through implementations. We conclude the paper with a discussion on computational privacy and other notions of privacy such as the non-unique retrieval of the private information from the protocol outputs