Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

Optimal spanners for axis-aligned rectangles

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the $L_1$-distance. The dilation of a pair of points is then defined as the length of the shortest rectilinear path between them that stays within the union of the rectangles and the connecting segments, divided by their $L_1$-distance. The dilation of the graph is the maximum dilation over all pairs of points in the union of the rectangles.We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain four results on this problem: (i) for arbitrary graphs, the problem is NP-hard; (ii) for trees, we can solve the problem by linear programming on $\mathrm{O}(n^2)$ variables and constraints; (iii) for paths, we can solve the problem in time $\mathrm{O}(n^3\log n)$; (iv) for rectangles sorted vertically along a path, the problem can be solved in $\mathrm{O}(n^2)$ time, and a $(1+\varepsilon )$-approximation can be computed in linear time.     

A Gauss-Seidel iteration method for nonlinear variational inequality problems over rectangles

This paper presents a simple yet practically useful Gauss-Seidel iterative method for solving a class of nonlinear variational inequality problems over rectangles and of nonlinear complementarity problems. This scheme is a nonlinear generalization of a robust iterative method for linear complementarity problems developed by Mangasarian. Global convergence is presented for problems with Z-functions. It is noted that the suggested method can be viewed as a specific case of a class of linear approximation methods studied by Pang and others.     

Packing random rectangles

A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0,1]. We prove that te number C _n of items in a maximum cardinality disjoint subset of n random rectangles satisfies

Double Aztec rectangles

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies et al. (1992) [2], [3]). Moreover, we consider the connection between the generating function and MacMahon's q-enumeration of plane partitions fitting in a given box     

Coloring axis-parallel rectangles

For every k and r, we construct a finite family of axis-parallel rectangles in the plane such that no matter how we color them with k colors, there exists a point covered by precisely r members of the family, all of which have the same color. For r=2, this answers a question of S. Smorodinsky [S. Smorodinsky, On the chromatic number of some geometric hypergraphs, SIAM J. Discrete Math. 21 (2007) 676-687].     

Simplest optimal cutting patterns for equal rectangles

This paper presents branch-and-bound algorithms that can guarantee the simplest optimal cutting patterns of equal rectangles. An existing linear algorithm determines the global upper bound exactly. The branching process ends when a branch of a lower bound equal to the global upper bound is found.     

A generalization of Riordan''s formula for 3xn latin rectangles

A formula is obtained for the number of three-line latin rectangles with first row normalized and second row a permutation which, in disjoint cycle form, has no cycles of prescribed lengths.     

An optimal contour algorithm for iso-oriented rectangles

Given a set of n iso-oriented rectangles in 2-space we describe an algorithm which determines the contour of their union in O(n log n + p) time and O(n + p) space, where p is the number of edges in the contour. This performance is time-optimal. The space requirements are the same as in the best previously known algorithm. We achieve this by introducing a new data structure, the contracted segment tree, which is a non-trivial modification of the well known segment tree. If only the pieces of the contour are to be reported then this approach yields a time- and space-optimal algorithm.     

A lower bound for n-diameters

Let D be a subset of the s-dimensional lattice Z^S, M=M(D) the number of elements in D, _Dthe space of trigonometric polynomials on the torus T^S with spectrum concentrated in D and having unit norm in L₂(T^S). In this paper we give the following bound for the Gel'fand diameter:d ⁿ( _D,C(T^s))M/2–N/2. This bound is subsequently used for actual functional classes.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 413–419, October, 1972.     

A bound for tridiagonal matrices

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 171–173, July–August, 1989.     

Mining maximal frequent rectangles

Advances in Data Analysis and Classification - Interval data, a special case of symbolic data, are becoming more frequent in different fields of applications due to the uncertainty in the...     

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

Optimal matrix-segmentation by rectangles

We study the problem of decomposing a nonnegative matrix into a nonnegative combination of 0-1-matrices whose ones form a rectangle such that the sum of the coefficients is minimal. We present for the case of two rows an easy algorithm that provides an optimal solution which is integral if the given matrix is integral. An additional integrality constraint makes the problem more difficult if the matrix has more than two rows. An algorithm that is based on the revised simplex method and uses only very few Gomory cuts yields exact integral solutions for integral matrices of reasonable size in a short time. For matrices of large dimension we propose a special greedy algorithm that provides sufficiently good results in numerical experiments.     

Symmetry-breaking constraints for packing identical rectangles within polyhedra

Two problems related to packing identical rectangles within a polyhedron are tackled in the present work. Rectangles are allowed to differ only by horizontal or vertical translations and possibly 90° rotations. The first considered problem consists in packing as many identical rectangles as possible within a given polyhedron, while the second problem consists in finding the smallest polyhedron of a given type that accommodates a fixed number of identical rectangles. Both problems are modeled as mixed integer programming problems. Symmetry-breaking constraints that facilitate the solution of the MIP models are introduced. Numerical results are presented.