Sides Of Rectangle Research Articles

The structure of optimal partitions of orthogonal polygons into fat rectangles

Motivated by a VLSI masking problem, we explore partitions of an orthogonal polygon of n vertices into isothetic rectangles that maximize the shortest rectangle side over all rectangles. Thus no rectangle is “thin”; all rectangles are “fat”. We show that such partitions have a rich structure, more complex than what one might at first expect. For example, for partitions all “cuts” of which are anchored on the boundary, sometimes cuts are needed 1/2 or 1/3 of the distance between two polygon edges, but they are never needed at fractions with a larger denominator. Partitions using cuts without any restrictions seem especially complicated, but we establish a limit on the “depth” of cuts (roughly, how distant from the boundary they “float” in the interior) and other structural constraints that lead to both an O( n) bound on the number of rectangles in an optimal partition, as well as a restriction of the cuts to a polynomial-sized grid. These constraints may be used to develop polynomial-time dynamic programming algorithms for finding optimal partitions under a variety of restrictions.

Numerical Investigation of the Second Transition in the Problem of Plane Convective Flow through a Porous Medium

The problem of convective flow through a porous medium in a plane rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. Single-parameter families of steady-state regimes resulting from the existence of cosymmetry of the corresponding differential equations are investigated using the Galerkin method. The onset and development of instability on these families and the characteristics of convective regimes as functions of the seepage Rayleigh number and the rectangle side ratio are studied. It is shown that the number of regimes which lose stability, the instability type, the number of convective rollers developed, and the heat transfer depend significantly on the vessel geometry. Several bifurcations of single-parameter families of steady-state regimes are identified and investigated.

The Buffon's needle problem and the design of a geophysical survey

A logical procedure for designing a geophysical survey when sampling an area with a regular grid can be summarized as follows: model the expected anomaly, estimate the expected noise level, estimate the area of the anomaly above the noise level, choose the spacing, in both the x‐ and y‐directions, of the measurement grid. This last step can be approached according to two main strategies: either when applying the sampling theorem to the shortest dimension of the anomaly or when using a coarser grid, leaving a more complete definition of the anomaly to a later fitting. When following this second option, it can be constructive to estimate the probability of intercepting a given anomaly with a specific segment of profile and a given profile spacing. This latter procedure is analysed by considering a rectangle approximating the plane projection of the anomaly shape and taking into account various ratios between the grid spacing and the rectangle sides. The formulae for estimating the probability of intersecting a given anomaly with a given segment of a given profile spacing are calculated. To demonstrate the accuracy of the results, a Monte‐Carlo simulation on a synthetic magnetic map was performed, obtaining, for different ratios between the sides, segment length and profile interval, an agreement better than 0.1% with the analytical formulae.

On Covering Z-Grid Points by Rectangles

Abstract A problem of combinatorial geometry is discussed: Cover a finite set of points lying on an integer grid in the Euclidean plane by regular rectangles such that the total area, circumference and number of rectangles used is minimized. This problem seems to be NP-hard, which is surely the case for related problems concerning covering points arbitrarily distributed in the plane. Treating the case of minimal rectangle side lengths k λ (grid constant), we propose an exact deterministic algorithm based on set theoretic dynamic programming, which then is improved by exploiting the rectangular and underlying grid structure. We also discuss a variant given by a further parameter p bounding the maximal possible covering cardinality. For this, we are able to find a time bound by a polynomial of degree O(p) . Generalizations to arbitrary values of k and arbitrary (finite) space dimensions are possible.

Laminar Forced Convection Modeling of Isothermal Rectangular Plates

A composite model for area-average Nusselt number for forced, laminar e ow parallel to a e nite, isothermal rectangular plate for a wide range of Reynolds numbers is proposed. The correlation equation is based on the superposition of the dimensionless shape factor and a modie ed laminar e ow boundary-layer asymptote with an empirically determined interpolation parameter. The Nusselt and Reynolds numbers and the dimensionless shape factor are based on either the rectangle side dimension parallel to the e ow direction or the square root of the heat transferarea.Theproposedcorrelation equationsareapplicableto rectangleswith sidedimensionratiosina range from 1 to 10. Extensive numerical results were used to e nd the optimal values of the interpolation parameter to provide close agreement between the correlation equation predictions and the numerical values. Nomenclature A = plate surface area, ´L £ W, m 2 C;CT = boundary-layer solution coefe cients F.Pr/ = Prandtl number function, Eq. (14) h = convective coefe cient, W/m 2 K

Perceptual causality in the amodal completion of kinetic structures.

Perceptual causality as an amodal completion of kinetic structures was demonstrated. The experimental situation was as follows: a square appeared on the left, moved from left to right till it touched one side of a stationary rectangle (a perceptual screen), and disappeared; after a while, another square appeared at the other side of the rectangle and moved right. The first square moved faster than the second one. Sometimes, two moving objects were perceived, with an amodal launching effect: the first moving object seemed to collide with the second one behind the screen. When both the screen and the occluding interval were small, despite the discrepancy of velocities, a single moving object was perceived, but the object seemed to abruptly change its speed behind the screen. The perception of an amodal completion of movement raises some theoretical issues: the discrepancy between amodal completion of kinetic structures and that of static ones, the problem of perceptual identity, and figure-ground segregation in dynamic scene. The dilemma of the naive subject will also be discussed.

The Maximum of a Random Walk and Its Application to Rectangle Packing

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Preparation of three-dimensionally oriented polycrystalline Si film

The crystal structure and morphology of polycrystalline Si films deposited using a plasma-enhanced chemical vapor deposition method, have been investigated by x-ray diffraction and atomic force microscopy. When the 〈110〉-oriented films were prepared at 690 °C with a hydrogen dilution ratio H2/SiH4=3 and rf power of 20 W, it was found that the grains with almost the same size are uniformly distributed within the film, and their shapes in a top view become rectangular. The longer sides of rectangles stand in line in the direction of gas flow (the plasma also spreads in this direction), indicating that the gas-flow direction and the plasma assist the growths of essentially three-dimensionally oriented grains.

The Productivity And Optimum Pattern Shape For Horizontal Wells Arranged In Staggered Rectangular Arrays

Abstract In a previous paper(1), the authors analyzed the productivity of horizontal wells arranged in a regular rectangular pattern. It was shown that there was an optimum for the pattern shape (i.e. length to width ratio); methods were described for calculating this shape and the productivity of the well. In this paper, the analysis has been extended to similar patterns in which the rows of wells are staggered. The optimum shape and performance have been calculated in a similar manner to that in the previous paper. It is found that the staggered pattern has superior characteristics. It has a slightly higher productivity and, more importantly, it is much less sensitive to the shape of the pattern. Essentially, the same performance is found over a wide range of pattern shapes for any particular drainage area. Equations are given for predicting the well productivity in this preferred arrangement as a function of well length and pattern area for pseudo-steady state flow. Introduction This paper is a continuation of a previously presented analysis(1) regarding the regular pattern of horizontal wells extended to the pattern in which the rows of wells are staggered. Figure 1 shows the distribution of wells in the reservoir with part la showing a regular pattern, and 1b a staggered pattern. In both cases, each well of length L is placed in the centre of a rectangle having dimensions b × a. In terms of geometry, both patterns are equivalent if only the respective dimensions are equal. In the previous paper(1), the authors presented a method which made it possible to find the optimum pattern shape and predict the productivity of the well in a two-dimensional reservoir with a modification permitting the productivity evaluation in a three-dimensional reservoir. A similar method will be used to extend the consideration to the staggered pattern. Description of the Problem The pattern shown in Figure 1b contains a repeated "pattern cell" that has been shaded. It can be seen that all boundaries of the shaded rectangle are the symmetry axes, thus being no-flaw boundaries. This means the liquid cannot flow across the sides of the rectangle, but can flow along them. Inside the rectangle, there is another no-flow boundary which is shown arbitrarily by the thick, dashed line, and whose exact location is not yet known. It must separate the drainage areas of both wells draining the rectangle. The shaded area in Figure 1b is, obviously twice as large as that in Figure la. The problem is treated as two-dimensional, pseudo-steady state flow and thus described by the equation Equation 1 (available in full paper) with boundary conditions: zero pressure gradient normal to the sides of the shaded rectangle in Figure 1b. With constant compressibility of the fluid and constant values of permeability and viscosity, equation(1) can be reduced to Poisson's equation [ef Ref. 1], and further replaced by a system of difference equations. The procedure is described in detail in the literature(1–3). Ultimately, the shaded rectangle is treated as a grid pattern containing mm grid points.

Indices of muscle fatigue

Myoelectric signal variables and mechanical variables are known to change during sustained voluntary or electrically elicited contractions. These phenomena reflect changes in the properties of the muscle fiber and its membrane. Such changes are generally called “fatigue.” Muscle fiber conduction velocity and myoelectric signal spectral parameters (mean and median frequency) show a linear or curvilinear decrease in time, depending on the level of voluntary or electrically elicited contraction. Amplitude paramaters [average rectified value (ARV) and root mean-square (RMS) value] and force often show a dome-shaped pattern with respect to time. In previous research, these patterns have been fitted with least-square regression curves or lines whose parameters (decrement, time constant, initial slope) have been considered indicators of the amount and rate of muscle property changes and therefore of ongoing fatigue. A new index of fatigue is proposed in this work. The product of a reference value (e.g., the first value of the time series) and the time of observation defines a reference rectangle. The area between the upper side of such rectangle and the experimental data points is divived by the area of the reference rectangle to provide this index. This area ratio index may be computed either as an attribute of a contraction or as a function of time; it is regression-free, it is dimensionless, it varies between 0 and 1 for decreasing patterns, it is negative for increasing patterns, and it is little affected by experimental point fluctuations except for the value that defines the reference rectangle. With respect to other indices, the area ratio index provides a quantitative approach to fatigue that is consistent with the intuitive definition of fatigue. If applied to different myoelectric signal variables, the resulting area ratios may be interpreted as the components of a fatigue vector.

One Area of Basic Misunderstanding among Teacher Candidates

Understanding the calculation of area is important to mathematics learning and real world applications and contributes to teachers' diagnosis and correction of students' errors. Because our experience led us to suspect that many teachers may lack adequate concepts of measurement of area, we tested 102 undergraduate (sophomore) students (46 men and 56 women) in an introductory education class at a state university. Each subject was given a test paper printed with instructions and a model rectangle marked to indicate it was 5 units wide x 6 units long. Four other similarly marked lines were printed below, each representing the left side of a new rectangle to be drawn by the subject so as to have an area equal to the model rectangle. These Lines, positioned vertically along the left margin to become the left sides of new rectangles, were 2, 3, 4 and 10 marked units long. Subjects were asked to draw the three missing sides of these new rectangles. This task was similar to one developed by Piaget and also used in a recent study of children's judgments of area (Hirstein, et al., 1978), except college-age adults were tested. Responses were correct if the missing sides were drawn to the correct lengths or if the appropriate unit marks were visible, even if actual proportions were distorted. Based on subjects' written explanations and drawings, errors were classified according to the misconceptions found in children's thinking by Hirstein, et 01. (1978). To identify possible influencing factors chi-squared values were calculated for performance on the area test and each of the following: number of high school mathematics courses, college GPA, year in college, sex, and teaching major. Only sex was statistically significant; men performed better than women (p < .01). Of the 102 students tested 57 (35 women and 19 men) or 56% did not draw the correct length of the missing side of one or more of the four new rectangles. Virtually all errors were classifiable as one of the misconceptions identified by Hirstein, et al. as common among children. Four (two men and two women) employed the mOSK primitive of misconceptions (constructing all new rectangles the same length as the original figure even though each had a different width) and 37 confused the concepts of perimeter and area: area is thought to be calculated by counting the units around the edge. Eighteen subjects saw the given units as indivisible rather than subdividable, that is, they were unable to deal successfully with the rectangle which required a fractional side. Only 44% (18 of 56 women and 27 of 46 men) were able to perform this simple task of measurement of area (drawing all four rectangles correctly), while 56% apparently were thinking about area measured in the same naive ways as many young children.

On the accoustic field of a point source in a rectangular space bounded by thin elastic walls: PMM vol. 43, no. 2, 1979, pp. 305–313

An analytic formula is obtained for Green's function of the Helmholtz equation in a rectangle. Along the rectangle sides boundary conditions containing higher order derivatives, and at its vertices boundary contact conditions are specified. Such boundary contact problem occurs in investigations of the acoustic field in a rectangular space bounded by thin elastic walls. The present paper is a continuation of papers [1, 2] where similar problems for the quarter-plane and half-band were considered.

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorDxDy (Dx andDy here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationDxDyf (x, y)=g (x, y, f, Dxf, Dx2f,DxDyf, Dy2f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).