Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

The Spin Holonomy Group In General Relativity

It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the time-independence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in the presence of a cosmological constant. It also shows that once matter is coupled to gravity, the "conservation of holonomy" is lost. When the fundamental group of space is non-trivial, the holonomy group need not be connected. For each homotopy class of loops, the holonomies comprise a coset of the full holonomy group modulo its connected component. These cosets are also time-independent. All possible holonomy groups that can arise are classified, and examples are given of connections with these holonomy groups. The classification of local and global solutions with given holonomy groups is discussed.Comment: 21 page

Multiobjective analysis for the design and control of an electromagnetic valve actuator

The electromagnetic valve actuator can deliver much improved fuel efficiency and reduced emissions in spark ignition (SI) engines owing to the potential for variable valve timing when compared with cam-operated, or conventional, variable valve strategies. The possibility exists to reduce pumping losses by throttle-free operation, along with closed-valve engine braking. However, further development is required to make the technology suitable for accept- ance into the mass production market. This paper investigates the application of multiobjective optimization techniques to the conflicting objective functions inherent in the operation of such a device. The techniques are utilized to derive the optimal force–displacement characteristic for the solenoid actuator, along with its controllability and dynamic/steady state performance

Proof of principle of a high-spatial-resolution, resonant-response gamma-ray detector for Gamma Resonance Absorption in 14N

The development of a mm-spatial-resolution, resonant-response detector based on a micrometric glass capillary array filled with liquid scintillator is described. This detector was developed for Gamma Resonance Absorption (GRA) in 14N. GRA is an automatic-decision radiographic screening technique that combines high radiation penetration (the probe is a 9.17 MeV gamma ray) with very good sensitivity and specificity to nitrogenous explosives. Detailed simulation of the detector response to electrons and protons generated by the 9.17 MeV gamma-rays was followed by a proof-of-principle experiment, using a mixed gamma-ray and neutron source. Towards this, a prototype capillary detector was assembled, including the associated filling and readout systems. Simulations and experimental results indeed show that proton tracks are distinguishable from electron tracks at relevant energies, on the basis of a criterion that combines track length and light intensity per unit length.Comment: 18 pages, 16 figure

Cooperating or Fighting with Decoherence in the Optimal Control of Quantum Dynamics

This paper explores the use of laboratory closed-loop learning control to either fight or cooperate with decoherence in the optimal manipulation of quantum dynamics. Simulations of the processes are performed in a Lindblad formulation on multilevel quantum systems strongly interacting with the environment without spontaneous emission. When seeking a high control yield it is possible to find fields that successfully fight with decoherence while attaining a good quality yield. When seeking modest control yields, fields can be found which are optimally shaped to cooperate with decoherence and thereby drive the dynamics more efficiently. In the latter regime when the control field and the decoherence strength are both weak, a theoretical foundation is established to describe how they cooperate with each other. In general, the results indicate that the population transfer objectives can be effectively met by appropriately either fighting or cooperating with decoherence