Original Rectangle Research Articles

The Redesigning of Transformers Fins in Terms of Heat Transfer Rates

Statistical and analytical research were carried out in order to optimize geometrical variables for organic convective heat flow from the Transformers tank, and several fin designs for transformers assembly were presented for geometrical optimization. The original rectangle, wavy, trapezoidal, as well as triangular fin geometries were used in the computational fluid dynamics research.

Corner Defected Multi-Resonance Ultra-Wideband Rectangular Patch Antenna

Objectives: The main objective of this article is to present a novel corner defected ultra wideband (UWB-3.1 to 10.6 GHz) rectangular patch with a partial ground plane to enhance the bandwidth with multi-resonance characteristics. By defecting the corners, the original rectangle patch will be converted into defected shape of proposed one to satisfy the objective. Methods: A FR-4 substrate with height (h) of 1.6 mm, dielectric constant (e r) of 4.3 and the loss tangent of 0.025 was used to develop the required antenna. The antenna was excited by a 50 Ohm microstrip line. This antenna was designed, simulated and analyzed by using MS–CST (Microwave Studio – Computer Simulation Technology) to attain the outcomes such as return loss (S11< -10 dB), voltage standing wave ratio (VSWR < 2), improved gain, Omni-directional radiation patterns, and enhanced bandwidth. Findings: It exhibits a fractional bandwidth (FBW) of 152.9 % between 2 GHz and 15 GHz. The antenna resonates at multiple frequencies owing to truncation of corners and produces resonant frequencies of 3.87 GHz and 11.36 GHz with the corresponding return losses of -26.03 dB and -34.87 dB respectively in the simulation point of view. The antenna was fabricated and tested. The measured and simulated results of the desired antenna were compared, and they show good agreement. Applications: The proposed antenna is suitable for UWB, WiMAX (Worldwide Interoperability for Microwave Access), C and X – band communications. Keywords: Ultrawideband; Rectangular patch; Bandwidth; Return loss; VSWR; Gain

The Precise Calculation about the Inertia Moment of Rectangular Section Beam

This paper involves in the bending deformation, the rectangular section beam precise calculation about the actual moment of inertia of neutral axis. Mainly through the analysis of correlation, deduced because of the influence of deformation of beam cross section from the original rectangle into trapezoidal, actually the section about what would happen to the actual moment of inertia of neutral axis changes, and related to accurate calculation results are obtained. At the same time regardless of the cross section shape change under the ideal state of comparing with the results of approximation, aiming to study the difference between the calculations of normal stress on the beam cross section. The results can be used to supplement the material mechanics related theoretical formula.

A Construction That Produces Wallis-Type Formulas

Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myerson, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.

A DOCUMENT RECTIFICATION APPROACH DEALING WITH BOTH PERSPECTIVE DISTORTION AND WARPING BASED ON TEXT FLOW CURVE FITTING

In this paper, we propose a coordinate transform model and document rectification process for book dewarping. This model can handle both perspective distortion and book surface warping problems. The main idea is to find the boundary of a region, which is a rectangle on a flat surface before warping and using the correspondence between the warped boundary and the original rectangle sides to map each pixel in the warped region into the rectangle. On a book page, two text lines and the left and right boundaries of a text region form a rectangle before warping. Therefore, we detect two text flow curves and text region boundaries on the left and right sides on a warped image and use the two curves and two line segments to do the transformation. The experimental results illustrate the effectiveness of our proposed book dewarping approach.

Mixing of wavefunctions in rectangular microwave billiards

A set-up is described allowing the automatic registration of wavefunctions of quasi-two-dimensional microwave billiards of arbitrary shape. Tests of the apparatus with rectangular shaped billiards showed that a precision of some percent in the wavefunction amplitudes can be obtained, as far as isolated resonances are considered. For the case of overlapping resonances, however, the measurement yields wavefunctions which are close to a symmetric and an antisymmetric linear combination of the original rectangle eigenfunctions. The cause for this at first sight surprising result is discussed.

A note on the largest empty rectangle problem

We consider the following problem: given a rectangle containingn points, find the largest perimeter subrectangle whose sides are parallel to those of the original rectangle, whose aspect ratio is below a given bound, and which does not contain any of the given points. Chazelle, Drysdale and Lee have studied a variant of this problem with areas as the quantity to be maximized. They gave anO(nlog3 n) algorithm for that problem. We adopt a similar divide-and-conquer approach and are able to use the simpler properties of the perimeter measure to obtain anO(nlog2 n) algorithm for our problem.

Computing the Largest Empty Rectangle

We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result [13] takes $O(N^2 )$ worst-case and $O(N\log ^2 N)$ expected time. This paper presents an $O(N\log ^3 N)$ time, $O(N\log N)$ space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Bentley [1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.

A method for calculating the shapes of peaks resulting from fragmentations of metastable ions in a mass spectrometer. I. Peak shapes arising from single valued kinetic energy releases

Mass-analysed ion kinetic energy (MIKE) spectra record the translational energy of all the daughter ions formed from a single parent ion in fragmentation reactions in the second field-free region of a reversed geometry double-focusing mass spectrometer. From the individual peak shapes in such spectra, the distribution of kinetic energy releases, n(T) , of each reaction can be determined. To glean the information from an experimental peak shape, it is first necessary to determine the exact peak shape that arises when only a discrete energy is released. Previous methods employed to calculate the expected peak shape for a discrete energy release are critically discussed and a new approach is described. The present method is based on the assumptions th at the aberrations of a cylindrical electrostatic analyser are not significant compared to its energy dispersion and th at the parent ion beam has a very small angular divergence. An analytical and a computer-based numerical solution have been developed, and they both predict that the peak shape for a single energy release is almost rectangular. In addition, the method has been extended to show how this shape changes when some of the ions are prevented from reaching the detector owing to the finite length of the collector slit. Again, both methods give identical forms for the resultant peak shape. The centre of the original rectangle becomes dished and the depth of the dishing increases in a predictable manner as either the slit length is reduced or the energy release is increased.

2. Notice of the remarkable Mathematical Properties of a certain Parallelogram

The parallelogram which formed the subject of this communication is the rectangle whose sides are to each other in the ratio of the diagonal of a square to its side,—a figure well known to architects, sculptors, and painters, from its beauty, and frequently adopted in the practical arts.The author shewed that if the given rectangle be bisected by a line parallel to its shortest side, each segment will be a figure similar in all respects to the original rectangle; and if either of these halves be itself bisected in the same manner, their halves will be rectangles similar to the original rectangle; and so on ad infinitum. The sides of the primary figure and its halves are continual proportionals, represented by the series The author endeavours to trace an analogy between the properties of this parallelogram and the logarithmic spiral.