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In the paper it is shown that there exists a function \\[U\\in L^1[0,1)^2\\], which is universal for all class \\[L^{p}[0,1)^2\\], \\[p\\in (0,1)\\], by rectangles and by spheres with respect to the double Walsh system in the sense of signs of Fourier coefficients.

Plimpton 322, which is a broken a clay tablet (cuneiform) from the Old Babylonian period (1900 - 1600 BCE), has been one of the very remarkable and enigmatic mathematical objects from antiquity. It features a table of 15 Pythagorean triples, involving many impressively large numbers. There has been much scholarly work, intense speculation, and theorizing, since the early work of Neugebauer and Sachs from 1945, on what the contents of the table signify. In the present article, Daniel F. Mansfield, after reviewing some of the present understanding on the issue, proposes a new explanation, on the theme that they concern a particular geometric problem in contemporary surveying, involving investigation of rectangles whose sides are \"regular\" numbers in the sense of Babylonian arithmetic. He notes that the study may have been motivated either entirely by a particular practical need or a theoretical interest in geometry, pursuing an issue.

Japanese rooms are measured by the number of tatami mats that will fit inside. The size of a tatami mat can vary by region, but is generally around 180 cm by 90 cm, giving it a 2:1 ratio of length to width. In the following, for simplicity, we suppose that each tatami mat is a rectangle with two adjacent sides of lengths 1 and 2. A typical tea ceremony room is square-shaped and its area is the equivalent of 4 and a half tatami mats. Questions regarding to lay tatami mats are not only fun for elementary school students, but also often included in entrance exams. In this paper, we derive recurrence formulae for determining the number of ways to lay tatami mats in a rectangular room whose vertical length is fixed at four or less, by using the concept of compartments or indivisible factors. Since the area of each tatami mat is two, if the area of the room is odd, only one half-sized tatami mat is allowed to be used. Therefore, if the vertical length of the room is three, the results will be different depending on whether the horizontal length of the room is even or odd. A generating function is used in this case, since it is difficult to derive the recurrence formula from direct consideration.

This study aims to describe the ethnomatematics study: square and rectangular on the motif of Malang batik artwork. This research is a qualitative-explorative research that is exploring batik motifs in square and rectangular shapes. The data in this study are in the form of library study data from both documents and electronic media. In addition, in the form of interview records related to Malang batik artwork and observations. The results of the study from this study indicate that: 1) on the kawung motif and Malang written batik there is a square concept; and 2) in Malang's masked batik motif there is a rectangular concept. The concept of square and rectangle in Malang batik artwork can be applied in mathematics learning. Especially in two-dimentional figure material.

The solar power tower technology has the advantages of green pollution-free and stable power generation, which has gradually attracted extensive attention from researchers in recent years. The quick computation of the solar heliostat field’s optical efficiency carries substantial potential for refining the design of the heliostat field, thereby enhancing the efficiency of power generation. Based on the parallel light hypothesis, we propose a fast algorithm designed to compute the optical efficiency of heliostat field. The algorithm takes into account various optical loss terms such as mirror reflectivity, cosine efficiency, shadowing and blocking efficiency, atmospheric transmittance efficiency and truncation efficiency. When calculating the shadowing and blocking efficiency, the “judgment rectangle” is used to quickly screen the candidate heliostat that may produce shadowing and blocking to the target heliostat. When calculating the truncation efficiency of the receiver surface, the strip discrete method and geometric projection method are used to quickly judge whether the reflected light is intercepted by the receiver, thus improving the calculation speed of optical efficiency. Consider a surrounding heliostat field as a test case, the proposed algorithm is compared with the conventional ray tracing, and the effectiveness and efficiency of the new algorithm are verified. The algorithm proposed in this paper has the advantages of accurate results and high calculation efficiency, and provides valuable insights that can be utilized for optimizing the design of heliostat field and improving the efficiency of tower solar thermal power generation.

As matrix representations of magic labelings of related hypergraphs, magic squares and their various variants have been applied to many domains. Among various subclasses, trimagic squares have been investigated for over a hundred years. The existence problem of trimagic squares with singly even orders and orders 16n has been solved completely. However, very little is known about the existence of trimagic squares with other even orders, except for only three examples and three families. We constructed normal trimagic squares by using product constructions, row–square magic rectangles, and trimagic pairs of orthogonal diagonal Latin squares. We gave a new product construction: for positive integers p, q, and r having the same parity, other than 1, 2, 3, or 6, if normal p × q and r × q row–square magic rectangles exist, then a normal trimagic square with order pqr exists. As its application, we constructed normal trimagic squares of orders 8q3 and 8pqr for all odd integers q not less than 7 and p, r ∈ 7, 11, 13, 17, 19, 23, 29, 31, 37. Our construction can easily be extended to construct multimagic squares.

We show that limt→0 eitΔ f(x) = f(x) almost everywhere for all f ∈ Hs(ℝ²) provided that s > ⅓. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

Grasping has always been a great challenge for robots due to its lack of the ability to well understand the perceived sensing data. In this work, we propose an end-to-end deep vision network model to predict possible good grasps from real-world images in real time. In order to accelerate the speed of the grasp detection, reference rectangles are designed to suggest potential grasp locations and then refined to indicate robotic grasps in the image. With the proposed model, the graspable scores for each location in the image and the corresponding predicted grasp rectangles can be obtained in real time at a rate of 80 frames per second on a graphic processing unit. The model is evaluated on a real robot-collected data set and different reference rectangle settings are compared to yield the best detection performance. The experimental results demonstrate that the proposed approach can assist the robot to learn the graspable part of the object from the image in a fast manner.

Ship object detection in synthetic aperture radar (SAR) images is both an important and challenging task. Previous methods based on horizontal bounding boxes struggle to accurately locate densely packed ships oriented in arbitrary directions, due to variations in scale, aspect ratio, and orientation, thereby requiring other forms of object representation, like rotated bounding boxes (OBBs). However, most deep learning-based OBB detection methods share a single-stage paradigm to improve detection speed, often at the expense of accuracy. In this paper, we propose a simple yet effective two-stage detector dubbed ORPSD, which enjoys good accuracy and efficiency owing to two key designs. First, we design a novel encoding scheme based on outer-rectangle projection (ORP) for the OrpRPN stage, which could efficiently generate high-quality oriented proposals. Second, we propose a convex quadrilateral rectification (CQR) method to rectify distorted shape proposals into rectangles by finding the outer rectangle based on the minimum area, ensuring correct proposal orientation. Comparative experiments on the challenging public benchmarks RSSDD and RSAR demonstrate the superiority of our ORPDet over previous OBB-based detectors in terms of both detection accuracy and efficiency.

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time O(nlog2n) for d=2 (Aggarwal and Suri 1987) and near nd for d≥3. We describe faster algorithms with the following running times (where ε>0 is an arbitrarily small constant and O~ hides polylogarithmic factors):n2O(log∗n)logn for d=2,O(n2.5+ε) for d=3, andO~(n(5d+2)/6) for any constant d≥4. To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee’s measure problem to optimize certain objective functions over the complement of a union of orthants.