Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Microwave Gaseous Discharges

Contains reports on two research projects

Microwave Gaseous Discharges

Contains reports on three research projects

Microwave Gaseous Discharges

Contains reports on three research projects.United States Atomic Energy Commission (Contract AT (30-1) 1842

How can I find what I want? Can children, chimpanzees and capuchin monkeys form abstract representations to guide their behavior in a sampling task?

Authors are grateful to the Royal Zoological Society of Scotland (RZSS) and the University of St Andrews for core financial support to the RZSS Edinburgh Zoo’s Living Links Research Centre, where this project was carried out. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. [639072]). We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number 2016-05552].Abstract concepts are a powerful tool for making wide-ranging predictions in new situations based on little experience. Whereas looking-time studies suggest an early emergence of this ability in human infancy, other paradigms like the relational match to sample task often fail to detect abstract concepts until late preschool years. Similarly, non-human animals show difficulties and often succeed only after long training regimes. Given the considerable influence of slight task modifications, the conclusiveness of these findings for the development and phylogenetic distribution of abstract reasoning is debated. Here, we tested the abilities of 3 to 5-year-old children, chimpanzees, and capuchin monkeys in a unified and more ecologically valid task design based on the concept of “overhypotheses” (Goodman, 1955). Participants sampled high- and low-valued items from containers that either each offered items of uniform value or a mix of high- and low-valued items. In a test situation, participants should switch away earlier from a container offering low-valued items when they learned that, in general, items within a container are of the same type, but should stay longer if they formed the overhypothesis that containers bear a mix of types. We compared each species' performance to the predictions of a probabilistic hierarchical Bayesian model forming overhypotheses at a first and second level of abstraction, adapted to each species' reward preferences. Children and, to a more limited extent, chimpanzees demonstrated their sensitivity to abstract patterns in the evidence. In contrast, capuchin monkeys did not exhibit conclusive evidence for the ability of abstract knowledge formation.Publisher PDFPeer reviewe

Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian

We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $K_C$-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $K_C$-orbits, which is a "quotient" of the resolution of the null fiber.Comment: 17 pages; completely revised and add reference

Microwave Gaseous Discharges

Contains reports on five research projects.United States Atomic Energy Commission (Contract AT(30-1) 1842

Shapes of free resolutions over a local ring

We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially and exposition has been streamline

Class and rank of differential modules

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a substitute for the length of a free complex--and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over noetherian commutative rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones Mathematica