Collection Of Rectangles Research Articles

Proses Transformasi Visual ke Simbolik Mahasiswa dalam Menyelesaikan Masalah Luas Daerah di Bawah Kurva

Mathematical representation is an idea interpretation form of solving mathematical problems. The process of mathematical representation is essential to pay attention to, especially in carrying out transformations between forms of representation, because it impacts the final solution obtained. This study describes the flow of students' thinking in carrying out the transformation process from visual to symbolic representation. This descriptive research was carried out with a qualitative approach. This research phase includes preparing representation tests and interview guidelines, giving tests to 48 students, conducting interviews with selected students based on the accuracy of the mathematical representations produced, and evaluation. The evaluation is carried out by reducing the results of the incorrect representation test, presenting visual and symbolic representation data and the transformation process, and concluding the transformation process. The result of the transformation process from visual to symbolic representation in solving the problem of the area under the curve is carried out with a flow of thought, namely translating the shape of the polygon area into a collection of rectangles, determining the width and length of the partition of the polygon area, and determining the area of the polygon area as a collection of rectangles. In determining the area as a collection of rectangles, there are two different processes, making a mathematical model in the form of sigma notation and addition. This study provides insight into the flow of thinking in changing between forms of mathematical representation in the hope that it can anticipate errors in solving the broad problem under the curve.

A method for constructing the trajectory for an unmanned aerial vehicle in a city

A synthesis method for the reference trajectory of an unmanned aerial vehicle that flies around obstacles of urban buildings in the horizontal and vertical planes is proposed. To solve this problem, for the first time a conformal mapping is constructed for a collection of rectangles that approximate the obstacles on the digital map of a terrain.

Automatic representation and reconstruction of DBM from LiDAR data using Recursive Minimum Bounding Rectangle

Three-dimensional building models are important for various applications, such as disaster management and urban planning. The development of laser scanning sensor technologies has resulted in many different approaches for efficient building model generation using LiDAR data. Despite this effort, generation of these models lacks economical and reliable techniques that fully exploit the advantage of LiDAR data. Therefore, this research aims to develop a framework for fully-automated building model generation by integrating data-driven and model-driven methods using LiDAR datasets.The building model generation starts by employing LiDAR data for building detection and approximate boundary determination. The generated building boundaries are then integrated into a model-based processing strategy because LiDAR derived planes show irregular boundaries due to the nature of LiDAR point acquisition. The focus of the research is generating models for the buildings with right-angled-corners, which can be described with a collection of rectangles under the assumption that the majority of the buildings in urban areas belong to this category. Therefore, by applying the Minimum Bounding Rectangle (MBR) algorithm recursively, the LiDAR boundaries are decomposed into sets of rectangles for further processing. At the same time, the quality of the MBRs is examined to verify that the buildings, from which the boundaries are generated, are buildings with right-angled-corners. The parameters that define the model primitives are adjusted through a model-based boundary fitting procedure using LiDAR boundaries. The level of details in the final Digital Building Model is based on the number of recursions during the MBR processing, which in turn are determined by the LiDAR point density. The model-based boundary fitting improves the quality of the generated boundaries and as seen in experimental results, the quality depends on the average LiDAR point spacing. This research thus develops an approach which not only automates the building model generation, but also achieves the best accuracy of the model while utilizing only LiDAR data.

A [formula omitted]-approximation for strip packing

We study strip packing, which is one of the most classical two-dimensional packing problems: given a collection of rectangles, the problem is to find a feasible orthogonal packing without rotations into a strip of width 1 and minimum height. In this paper we present an approximation algorithm for the strip packing problem with absolute approximation ratio of 5/3+ε for any ε>0. This result significantly narrows the gap between the best known upper bound and the lower bound of 3/2; previously, the best upper bound was 1.9396 due to Harren and van Stee.

Computational Analysis and Efficient Algorithms for Micro and Macro OFDMA Downlink Scheduling

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Orthogonal frequency-division multiple access (OFDMA) is one of the most important modulation and access methods for the future mobile networks. Before transmitting a frame on the downlink, an OFDMA base station has to invoke an algorithm that determines which of the pending packets will be transmitted, what modulation should be used for each of them, and how to construct the complex OFDMA frame matrix as a collection of rectangles that fit into a single matrix with fixed dimensions. We propose efficient algorithms, with performance guarantee, that solve this intricate OFDMA scheduling problem by breaking it down into two subproblems, referred to as macro and micro scheduling. We analyze the computational complexity of these subproblems and develop efficient algorithms for solving them. </para>

$L^p$ estimates for maximal averages along one-variable vector fields in ${\mathbf R} ^2$

We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the unit square such that v(x, y) = (1, u(x)) for some measurable u : [0,1] → [0,1]. Let δ be a small parameter, and let R be the collection of rectangles R of a fixed width such that δ much of the vector field inside R is pointed in (approximately) the same direction as R. We show that the operator defined by (0.1) M R f(z) = sup 1 |R| f R |f| z∈R∈R is bounded on L P for p > 1 with constants comparable to 1 δ.

Two-dimensional bin packing with one-dimensional resource augmentation

The two-dimensional bin packing problem is a generalization of the classical bin packing problem and is defined as follows. Given a collection of rectangles specified by their width and height, pack these into a minimum number of square bins of unit size. Recently, the problem was proved to be APX-hard even in the asymptotic case, i.e. when the optimum solutions require a large number of bins [N. Bansal, J. Correa, C. Kenyon, M. Sviridenko, Bin packing in multiple dimensions: Inapproximability results and approximation schemes, Math. Oper. Res. 31 (1) (2006) 31–49]. On the positive side, there exists a polynomial time algorithm that uses OPT bins whose sides have length ( 1 + ϵ ) , where OPT denotes the number of unit sized bins used by the optimum solution [N. Bansal, J. Correa, C. Kenyon, M. Sviridenko, Bin packing in multiple dimensions: Inapproximability results and approximation schemes, Math. Oper. Res. 31 (1) (2006) 31–49]. A natural question that remains is the approximability of the problem when we are allowed to relax the size of the unit bins in only one dimension. In this paper, we show that there exists an asymptotic polynomial time approximation scheme for packing rectangles into bins of size 1 × ( 1 + ϵ ) .

Expected wasted space of optimal simple rectangle packing

A simple packing of a collection of rectangles contained in [0,1]2 is a disjoint subcollection such that each vertical line meets at most one rectangle of the packing. The wasted space of the packing is the surface of the area of the part of [0,1]2 not covered by the packing. We prove that for a certain number L, for all N≥2, the wasted space WN in an optimal simple packing of N independent uniformly distributed rectangles satisfies

Hinted quad trees for VLSI geometry DRC based on efficient searching for neighbors

Design-rule checking, whose efficiency depends greatly on the speed in finding an object's neighbors, is an indispensable component of any VLSI design process. Given a design, the objects it contains are usually represented by their smallest enclosing rectangles, and neighbor search is defined as the operation to find, among the collection of rectangles, the ones that are within the specified distance of a rectangle. Commonly, the rectangles are stored in a tree structure, and a region query that searches the tree starting at its root is used to find a rectangle's neighbors. In this paper, we introduce the hinted quad tree, or HQT, that supports neighbor searches directly without always starting a search at the root of the tree. We show that HQT achieves the highest neighbor-search performance among the data structures compared and uses a reasonable amount of storage.

A unified approach to geometric problems using dual cone transformation:

This paper uses a new transformation technique that allows a unified treatment to a number of geometric problems. In particular, we are able to offer an optimal algorithm for range query and point enclosure problems. More specifically, we discuss how our approach yields efficient algorithms to the following three classical problems in computational geometry: (1) Given a collection of d-rectangles in E d , how many of them intersect an arbitrary query rectangle. (2) Give a collection of rectangles which of them contain an arbitrary query point. (3) Given a set of n points in R d and a query d-range q, report the points of S that lie within q. All these results represent significant improvements over the best methods previously known.

Tiling the unit square with squares and rectangles

Call a rectangle small if it will fit inside the unit square; call a rectangle binary if its dimensions are powers of 2. • Theorem 1. If S is a set of small binary squares having total area greater than 1, some finite subset of S will tile the unit square . • Theorem 2. If R is a set of small binary rectangles having total area greater than 5 2 , some finite subset of R will tile the unit square . The above bounds are sharp. Various corollaries are drawn. In particular we show that if R is a collection of rectangles, then R can cover the plane iff Σ{min( a, 1) · min( b, 1) | R ϵ, R }; = ∞, where for a given rectangle, a is its width and b is its length.

General polygonal variogram functions: Evaluation of estimation variance integrals

The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a general polygonal function the requisite integrals may be easily computed by a closed form representation of simple integrals. This paper provides the integration formulas for two-dimensional variogram functions whose domain is represented as a finite collection of rectangles. The integration formulas essential for a fully developed polygonal approach to an extensive statistical evaluation of geostatistical quantities are presented.