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 three research projects

Microwave Gaseous Discharges

Contains reports on two research projects

Microwave Gaseous Discharges

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

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

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

Microwave Gaseous Discharges

Contains research objectives and reports on five research projects

Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Anatomical localization of progenitor cells in human breast tissue reveals enrichment of uncommitted cells within immature lobules

Introduction: Lineage tracing studies in mice have revealed the localization and existence of lineage-restricted mammary epithelial progenitor cells that functionally contribute to expansive growth during puberty and differentiation during pregnancy. However, extensive anatomical differences between mouse and human mammary tissues preclude the direct translation of rodent findings to the human breast. Therefore, here we characterize the mammary progenitor cell hierarchy and identify the anatomic location of progenitor cells within human breast tissues. Methods: Mammary epithelial cells (MECs) were isolated from disease-free reduction mammoplasty tissues and assayed for stem/progenitor activity in vitro and in vivo. MECs were sorted and evaluated for growth on collagen and expression of lineages markers. Breast lobules were microdissected and individually characterized based on lineage markers and steroid receptor expression to identify the anatomic location of progenitor cells. Spanning-tree progression analysis of density-normalized events (SPADE) was used to identify the cellular hierarchy of MECs within lobules from high-dimensional cytometry data. Results: Integrating multiple assays for progenitor activity, we identified the presence of luminal alveolar and basal ductal progenitors. Further, we show that Type I lobules of the human breast were the least mature, demonstrating an unrestricted pattern of expression of luminal and basal lineage markers. Consistent with this, SPADE analysis revealed that immature lobules were enriched for basal progenitor cells, while mature lobules consisted of increased hierarchal complexity of cells within the luminal lineages. Conclusions: These results reveal underlying differences in the human breast epithelial hierarchy and suggest that with increasing glandular maturity, the epithelial hierarchy also becomes more complex. Electronic supplementary material The online version of this article (doi:10.1186/s13058-014-0453-3) contains supplementary material, which is available to authorized users