Congruent Squares Research Articles

Local refinement with arbitrary irregular meshes and implementation in numerical manifold method

The solution accuracy based on finite element interpolation strongly depends on mesh quality. Thus, the numerical manifold method advocates that if a finite element mesh is selected as the mathematical cover, the mesh should be uniform, where all elements are congruent squares or isosceles right triangles for two-dimensional problems. However, the elements close to the points of stress concentrations or singularities must be smaller than those further away from these points. This leads to an irregular mesh in which a few nodes of certain small elements are hung on the edges of adjacent larger elements. This study implements local refinement with arbitrary k-irregular meshes by selecting piecewise polynomials as the local approximations over physical patches and enforcing appropriate constraints on the piecewise polynomials. All the elements are ordinary isoparametric elements, and elements with variable numbers of nodes on edges are not required. No linear dependency issue exists. Furthermore, refinement with any number of mesh layers is developed so that solution error can be distributed as uniformly as possible. Typical examples with different singularities are designed to illustrate the effectiveness of the proposed procedure.

Wafer Defect Inspection Optimization With Partial Coverage—A Numerical Approach

Electron beam inspection (EBI) with high resolution is a promising technique to improve the defect inspection on the surface of patterned wafer. However, high resolution usually means long inspection time, which results in the low throughput and limitation of EBI applied in practice. This study aims to optimize the inspection time of EBI by reducing the total number of inspection regions without loss of the accuracy. We first refine this defect inspection optimization problem as a partial congruent square cover problem. Then, we propose two novel mixed-integer linear programming models for this problem. To deal with the large-scale problems, an approximation algorithm is developed to obtain the high-quality solutions. This approximation algorithm efficiently utilizes the linear programming (LP) rounding technique and greedy strategy based on the proposed model. Compared with the existing algorithms in the literature, numerical results show the superiority of the proposed model and algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Defect inspection is a key process in wafer fabrication for identifying and inspecting the patterning defects generated during the complicated fabrication processes. Electron beam inspection (EBI) takes place of optical inspection gradually as the design rules keep shrinking and the circuits are more susceptible to nanoscale killer defects. Low throughput is the main drawback of EBI and the improvements on throughput have far-reaching significance on the high volume manufacturing of semiconductor products. This study aims to reduce the number of inspection regions to improve the total inspection time in the EBI process. Considering that the inspection time for each inspection region is constant, less inspection regions means less total inspection time. However, the positions of inspection regions are arbitrary across the continuous planar space, putting pressure on modeling and solving the problem. A preprocessing algorithm is designed to discover a limited number of candidate positions of inspection regions without loss of optimality, which greatly simplifies the problem. For dealing with large-scale instances, an approximation algorithm combining linear programming (LP)-rounding technique and greedy strategy is designed to get near optimal solutions. Since the number of inspection regions is one key factor determining the total inspection time, the proposed optimization methods have a potential to be applicable to enhance the efficiency of advanced EBI platforms, such as ASML HMI eP series.

Consistent dynamic map labeling with fairness and importance

Geographical visualization systems, such as online maps, provide interactive operations of continuous zooming and panning. With consistent dynamic map labeling, users can navigate continuously in the map areas such that labels are not allowed to exhibit abrupt change in terms of their positions or sizes, and labels should not suddenly disappear or reappear when zooming in or pop up when zooming out. However, existing works on consistent dynamic map labeling only study the problem of optimizing overall visibility of labels across all zooming scales, and overlook the fairness and importance issues when labels are selected for showing at different scales. Such issues are in fact important ones, and need to be addressed. In this paper, we propose to assign weights to labels to reveal their corresponding importance and study the novel problem of maximizing minimum weighted active range (MMWAR), in which we compute an active range assignment such that (1) no two active ranges overlap and (2) the minimum weighted active range height is maximized. In particular, we study the simple MMWAR problem by restricting the active ranges such that a label is never deselected when zooming in, and present optimal O(n2log⁡n) time algorithms for 1D and 2D cases, where n is the number of labels. We present an O(nlog⁡n) time algorithm for simple MMWAR in 1D of congruent case. Further, we generalize the simple MMWAR problem to the ex-simple MMWAR problem, in which the starting active scale is not fixed to be zero anymore. On the problem complexity side, we prove that the ex-simple MMWAR problem in 2D is NP-hard, even if all extrusions are congruent square pyramids. Next, we propose two fast algorithms for the ex-simple MMWAR problem in which all labels have the same weight. Then, we present an Integer Linear Programming (ILP) formulation for computing the optimal solutions for several variants of the MMWAR problem. Finally, we evaluate the performance of our proposed algorithms on real world dataset experimentally.

Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares

This paper discusses the problem of covering and hitting a set of line segments L in R2 by a pair of axis-parallel congruent squares of minimum size. We also discuss the restricted version of covering, where each line segment in L is to be covered completely by at least one square. The proposed algorithms assume that the input segments are given in a read-only array. For each of these problems (i.e. covering, hitting and restricted covering problems), our proposed algorithm reports the optimum result by executing only two passes of reading the input array sequentially. All these algorithms need only O(1) extra space. The solution of these problems also give a 2 approximation for covering and hitting those line segments L by two congruent disks of minimum radius with same computational complexity.

Approximation algorithms on consistent dynamic map labeling

We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every scale and the sum of total visible ranges is maximized. This is also called the active range optimization (ARO) problem defined by Been et al. (2006) [2]. We propose approximation algorithms for several variants of this problem. For the simple ARO problem, we provide a 3clog⁡n-approximation algorithm for unit-width rectangular labels if there is a c-approximation algorithm for the unit-width label placement problem in the plane; and a randomized polynomial-time O(log⁡nlog⁡log⁡n)-approximation algorithm for arbitrary rectangular labels. For the general ARO problem, we prove that it remains NP-complete even for congruent square labels with equal selectable scale range. Moreover, we contribute 12-approximation algorithms for both arbitrary square labels and unit-width rectangular labels, and a 6-approximation algorithm for congruent square labels, and show that the bounds are tight.

A discrete square global grid system based on the parallels plane projection

We developed a direct partitioning method to construct a seamless discrete global grid system (DGGS) with any resolution based on a two‐dimensional projected plane and the earth ellipsoid. This DGGS is composed of congruent square grids over the projected plane and irregular ellipsoidal quadrilaterals on the ellipsoidal surface. A new equal area projection named the parallels plane (PP) projection derived from the expansion of the central meridian and parallels has been employed to perform the transformation between the planar squares and the corresponding ellipsoidal grids. The horizontal sides of the grids are parts of the parallel circles and the vertical sides are complex ellipsoidal curves, which can be obtained by the inverse expression of the PP projection. The partition strategies, transformation equations, geometric characteristics and distortions for this DGGS have been discussed. Our analysis proves that the DGGS is area‐preserving while length distortions only occur on the vertical sides off the central meridian. Angular and length distortions positively correlate to the increase in latitudes and the spanning of longitudes away from a chosen central meridian. This direct partition only generates a small number of broken grids that can be treated individually.

Families of irreptiles

The search for tilings of the plane by congruent images of some given polygon A leads in a natural way to the concept of a reptile. A is called a reptile if it can be dissected into finitely many pairwise congruent pieces which are similar images of A. We speak of a dissection if the covering pieces can only have boundary points in common. Many known examples of reptiles are polyominoes or polyiamonds. A polygon is called a polyomino (polyiamond) if it has a connected interior and possesses an edge-to-edge dissection into finitely many congruent squares (equilateral triangles). A family of reptiles that are polyominoes can be obtained as follows (see [2], [3, p. 97], [6, p. 54], and the first illustration in Fig. 1). Fix an integer k ≥ 1 and dissect a square S into (2k)2 congruent smaller squares S1, . . . , S(2k)2 . Let δc denote the rotation about the centre c of S by an angle of π2 . Now choose a simple polygonal arc contained in the union of the boundaries of the pieces Si that connects c with a point on the boundary of S such that ∩ δc( ) = {c}. Then , δc( ), and a quarter of the boundary of S bound a reptile A ⊆ S (shaded in Fig. 1). Indeed, since A splits into k2 congruent squares Si and since S as well as any other square admits a dissection into four congruent similar copies of A, A can be

Improved Dense Packings of Congruent Squares in a Square

Let sn be the side of the smallest square into which it is possible to pack n congruent squares. In this paper we link sn to the supremum of the maximal inflation Ω (C) of admissible configurations C. The computation and the properties of Ω in a bounded domain. We improve the best known packings of n equal squares for n=11, 29 and 37, and give an alternative optimal packing of 18 squares.

A Problem on Hinged Dissections with Colours

We examine the following problem. Given a square C we want a hinged dissection of C into congruent squares and a colouring of the edges of these smaller squares with k colours such that we can transform the original square into another with its perimeter coloured with colour i, for all i in *. We have the restriction that the moves have to be realizable in the plane, so when swinging the pieces no overlappings are allowed. We show a solution for k colours that uses p2 pieces, with p an even number and at least * this by using a necklace made of the p2 pieces and an ingenious way to wrap it into a square.

Teacher to Teacher: The Pentomino Square Problem

During part of a presentation to some middle school mathematics students, right before a vacation from school, we looked at pentominoes. (Pentominoes are shapes made from five congruent squares attached to each other along full sides.) See figure 1's illustration of the definition; two examples are pentominoes, one is not.

Lower Bounds on Strip Discrepancy for Nonatomic Colorings

A method of proof is given for obtaining lower bounds on strip discrepancy when the distributions do not have atoms. Partition the unit square into an chessboard of congruent square pixels, where n is even. Color of the pixels red, and the rest blue. For any convex set A, let be the difference between the amounts of red and blue areas in A. Under a technical local balance condition, we prove there must be a strip S, of width less than , for which , where c is a positive constant, independent of n and the coloring. The proof extends methods discovered by Alexander and further developed by Chazelle, Matousek, and Sharir. Integral geometric notions figure prominently.

ON PIERCING SETS OF AXIS-PARALLEL RECTANGLES AND RINGS

We consider the p-piercing problem for axis-parallel rectangles. We are given a collection of axis-parallel rectangles in the plane and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e, a set of p points as above) for values of p=1,2,3,4,5. The result for 4 and 5-piercing improves an existing result of O(n log 3 n) and O(n log 4 n) to O(n log n) time. The result for 5-piercing can be applied find an O(n log 2 n) time algorithm for planar rectilinear 5-center problem, in which we are given a set S of n points in the pane, and wish to find 5 axis-parallel congruent squares of smallest possible size whose union covers S. We improve the existing algorithm for general (but fixed) p to O(np-4 log n) running time, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.

Arcs with Positive Measure and a Space-Filling Curve

In this note we construct a one-parameter family of arcs in R2 with positive two-dimensional Lebesgue measure, which converge uniformly to a space-filling curve. By an arc we mean a homeomorphism of I = [0, 1] into I x I. The existence of such maps is not news, but our procedure for constructing them is so transparently simple that it can be presented in undergraduate or graduate courses in advanced calculus, real analysis, or topology. Furthermore, each of these curves is the uniform limit of a recursively defined sequence of very simple functions. The examples of space filling curves of Peano or Hilbert, which are generally cited (e.g., [1], [21, or [3]), are also constructed recursively, but the recursions are more complicated. All of them pass from one stage to the next by splicing together reduced and rotated copies of an initial figure. In our construction, no rotation is necessary to get them to fit, so the recursion has a much simpler description. Our construction is similar to one found in Section 10.10 of [1]. This work was motivated by problem E 2975, posed by F. B. Jones in the December, 1982, issue of the American Mathematical Monthly. We wish to thank the referee for several helpful suggestions. Let us first describe a typical member of our family of arcs intuitively. Imagine a necklace composed of four congruent squares and three connecting line segments arranged as in FIGURE 1. In each square we put another necklace, but one in which the + shaped region is proportionately much thinner. By leaving the original

Solid Polyomino Constructions

A plane polyomino of order n is defined by Solomon W. Golomb [4] as a set of n congruent squares that are simply connected, edge-to-edge. A three-dimensional, or solid, polyomino of order n is a set of n congruent cubes that are simply connected, face-to-face. These are easily made by gluing like cubes together in all permissible patterns. No formula has been discovered which gives the number of different polyominoes of order n, plane or solid. The solid ones of orders one to five are shown in FIGURE 1, each being identified by a number. The set contains 41 pieces and a total of 186 individual or unit cubes. Martin Gardner has discussed polyominoes in his Mathematical Games section of Scientific American and also in each of his three books [1, 2, 3]. Chapters thirteen in [1] and [3] are concerned with plane polyominoes. Chapter six in [2] deals with the well-known Soma cube of Piet Hein consisting of the pieces numbered 4, 6, 7 and 9-12 in FIGURE 1. Gardner points out that there are more than 230 essentially different ways of stacking these pieces in a 3 x 3 x 3 cube (or simply, a 3-cube) but to date no one knows precisely how many. A similar cube occurs in Steinhaus [5, p. 168] where pieces numbered 6, 11, 12, 30, 33 and 40 are used. It is easy to prove that there are just two solutions to the Steinhaus cube. The 30 and the 40 go together successfully in just one way. Then the 33 has only two possible positions after each of which the positions of 6, 11 and 12 are uniquely determined. Steinhaus remarks: Two pieces are congruent by symmetry. (Was it possible to avoid it?). He refers to numbers 11 and 12. We do not know his answer but it is certainly possible to avoid using either 11 or 12. Indeed, a 3-cube can be constructed using, in addition to the four polyominoes 6, 30, 33, 40, any of the pairs 8, 11 or 8, 12 or 9, 11 or 9, 12 or, for that matter, Y T1Z~~~~~ZZ~~~IflT 1 2 3 4 5

Models of specifically paired like (homologous) nucleic acid structures

A specific pairing of like Watson-Crick base pairs, and of structures containing them, might occur in which the pairs are related by dyad axes. If base pairs are related in this way the glycosidic bonds lie at the corners of figures which approximate to congruent squares. Models involving this pairing in which two Watson-Crick-Wilkins double helices are related by a dyad axis parallel to the long molecular axis can be built.