Suppose x̄ and s̄ lie in the interiors of a cone K and its dual K∗ respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x̄ and s̄ and incribed in K and K∗ respectively is maximized. Here the balls are defined using the two dual norms. We provide a solution when the cones are symmetric, that is self-dual and homogeneous. This prov...

We report high resolution (R∼37,000) integral field spectroscopy of the central region (r<14 arcsec) of the Red Rectangle nebula surrounding HD44179. The observations focus on the λ5800Å emission feature, the bluest of the yellow/red emission bands in the Red Rectangle. We propose that the emission feature, widely believed to be a molecular emission band, is not a molecular rotation contour, bu...

The rectangle packing problem often appears in encasement and cutting as well as very large-scale integration design. To solve this problem, many algorithms such as genetic algorithm, simulated annealing and other heuristic algorithms have been proposed. In this paper, a new heuristic algorithm is recommended based on two important concepts, namely, the corner-occupying action and caving degree...

Magic rectangles are well-known for its very interesting and entertaining combinatorics. In a magic rectangle, the numbers 1 to mn are arranged in an array of m rows and n columns so that each row adds to the same total M and each column to the same total N . In the present paper we provide a new systematic method for constructing any even by even magic rectangle. The method proposed is extreme...

The problem that we consider is the following: given an n × n array A of positive numbers, find a tiling using at most p rectangles (which means that each array element must be covered by some rectangle and no two rectangles must overlap) that minimizes the maximum weight of any rectangle (the weight of a rectangle is the sum of elements which are covered by it). We prove that it is NP-hard to ...

We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R1 \ R2 is connected for every pair of rectangles R1, R2 ∈ R. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of (1.5 + ...

In this paper, we propose the rectangle transformation problem (RTP) and its variants. RTP asks for a transformation by a rectangle partition between two rectangles of the same area. We are interested in the minimum RTP which requires to minimize the partition size. We mainly focus on the strict rectangle transformation problem (SRTP) in which rotation is not allowed in transforming. We show th...

In this paper we consider the problem of drawing a graph in the plane so that the vertices of the graph are drawn as rectangles and the edges are horizontal or vertical line segments. We are in particular interested in drawings where each of the line segments can be thickened to have positive width, and none of these thickened segments (which we call bands of visibility) intersects the interior...