Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure

Meteorite cloudy zone formation as a quantitative indicator of paleomagnetic field intensities and cooling rates on planetesimals

Metallic microstructures in slowly-cooled iron-rich meteorites reflect the thermal and magnetic histories of their parent planetesimals. Of particular interest is the cloudy zone, a nanoscale intergrowth of Ni-rich islands within a Ni-poor matrix that forms below 350{\deg}C by spinodal decomposition. The sizes of the islands have long been recognized as reflecting the low-temperature cooling rates of meteorite parent bodies. However, a model capable of providing quantitative cooling rate estimates from island sizes has been lacking. Moreover, these islands are also capable of preserving a record of the ambient magnetic field as they grew, but some of the key physical parameters required for recovering reliable paleointensity estimates from magnetic measurements of these islands have been poorly constrained. To address both of these issues, we present a numerical model of the structural and compositional evolution of the cloudy zone as a function of cooling rate and local composition. Our model produces island sizes that are consistent with present-day measured sizes. This model enables a substantial improvement in the calibration of paleointensity estimates and associated uncertainties. In particular, we can now accurately quantify the statistical uncertainty associated with the finite number of islands and the uncertainty on their size at the time of the record. We use this new understanding to revisit paleointensities from previous pioneering paleomagnetic studies of cloudy zones. We show that these could have been overestimated but nevertheless still require substantial magnetic fields to have been present on their parent bodies. Our model also allows us to estimate absolute cooling rates for meteorites that cooled slower than 10000{\deg}C My-1. We demonstrate how these cooling rate estimates can uniquely constrain the low-temperature thermal history of meteorite parent bodies.Comment: Manuscript resubmitted after revision

Connectivity Explains Local Ant Community Structure in A Neotropical Forest Canopy: A Large‐Scale Experimental Approach

Understanding how habitat structure and resource availability affect local species distributions is a key goal of community ecology. Where habitats occur as a mosaic, variation in connectivity among patches influences both local species richness and composition, and connectivity is a key conservation concern in fragmented landscapes. Similarly, availability of limiting resources frequently determines species coexistence or exclusion. For primarily cursorial arthropods like ants, gaps between neighboring trees are a significant barrier to movement through the forest canopy. Competition for limited resources such as nest sites also promotes antagonistic interactions. Lianas (woody vines) connect normally isolated neighboring tree crowns and often have hollow stems inhabited by ants. We used two large‐scale liana‐removal experiments to determine how connectivity and nest site availability provided by lianas affect arboreal ant species richness, species composition, and β‐diversity in a lowland tropical forest in Panama. Removing lianas from a tree crown reduced ant species richness up to 35%, and disproportionately affected species that require large foraging areas. Adding artificial connectivity to trees mitigated the effects of liana removal. Ant colonization of artificial nests was higher (73% occupied) in trees without lianas vs. trees with lianas (28% occupied). However, artificial nests typically were colonized by existing polydomous, resident ant species. As a result, nest addition did not affect ant community structure. Collectively, these results indicate that lianas are important to the maintenance of arboreal ant diversity specifically by providing connectivity among neighboring tree crowns. Anticipated increases in liana abundance in this forest could increase the local (tree‐level) species richness of arboreal ants, with a compositional bias toward elevating the density of broad‐ranging specialist predators

Classical Rules in Quantum Games

We consider two aspects of quantum game theory: the extent to which the quantum solution solves the original classical game, and to what extent the new solution can be obtained in a classical model.Comment: The previous title, "Quantum games are no fun (yet)", was too whimsical for Physical Review. This is a comment on most, but not all, papers on quantum game theor

Parameter space metric for 3.5 post-Newtonian gravitational-waves from compact binary inspirals

We derive the metric on the parameter space of 3.5 post-Newtonian (3.5PN) stationary phase compact binary inspiral waveforms for a single detector, neglecting spin, eccentricity, and finite-body effects. We demonstrate that this leads to better template placement than the current practice of using the 2PN metric to place 3.5PN templates: The recovered event rate is improved by about 10% at a cost of nearly doubling the number of templates. The cross-correlations between mass parameters are also more accurate, which will result in better coincidence tests.Comment: 10 pages, 7 figure