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"Rectangles"
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Universal functions for classes \\L^p0,1)^2\\ , \\pın (0,1)\\ , with respect to the double Walsh system
In the paper it is shown that there exists a function \\[Uın L^1[0,1)^2\\], which is universal for all class \\[L^p[0,1)^2\\], \\[pın (0,1)\\], by rectangles and by spheres with respect to the double Walsh system in the sense of signs of Fourier coefficients.
Journal Article
Rectangles and triangles : a song about drawing with shapes
Sing along while drawing an adventure with princesses, knights, and dragons, using rectangles and other shapes.
Book
Plimpton 322: a study of rectangles
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.
Journal Article
Counting The Number of Arrangements of Tatami Mats in a Rectangular Room of Vertical Length 2, 3 and 4
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.
Journal Article
Two Dimensional Object in square and rectangles: Batik artwork approach
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.
Journal Article
A note on piercing discrete rectangles
In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in \\( R^d\\). Very recently, this result was extended to the \\((p,q)\\) setting with \\(p q d+1\\) by Edwards and Soberón, and subsequently to the case \\(p q 2\\) by Gangopadhyay, Polyanskii, and the author of this paper. In this paper, we obtain improved bounds for the \\((p,q)\\) problem in the case \\(q=2\\) and \\(d=2\\). More precisely, our main result asserts that for any integer \\(p 2\\), any set \\(P R^2\\), and any finite family \\( B\\) of axis-parallel rectangles in \\( R^2\\) such that every rectangle contains a point of \\(P\\), if among every \\(p\\) rectangles there exist two whose intersection contains a point of \\(P\\), then there exists a subset \\(S P\\) of size at most \\(O\\!( (p p)^2 )\\) such that every rectangle contains a point of \\(S\\). Moreover, when \\(p=2\\), the size of \\(S\\) can be bounded by \\(4\\).
Paper
Nonlinear parabolic thin sets and parabolic Wolff inequalities
We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling \\(_c(x,t)=(cx,c^2t)\\) and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic \\(( q)\\)-thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of \\(( 2)\\)-thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points \\(z_0ın\\) for the heat operator \\(_t-\\) and for the degenerate operator \\(La=_t(|y|^a)-div(|y|^a)\\) in \\(^d+1\\) are negligible with respect to the thermal capacity \\(cap^ T\\) and the parabolic Bessel capacity \\(C_ 2\\), respectively.
Paper
A Note on One-Hole Domino Tilings of Squares and Rectangles
We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the \\((2k+1) (2k+1)\\) square grid, we show that the number of near-perfect matchings is a multiple of \\(2^k\\), and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of \\(2^k\\). We also determine the parity of the number of near-perfect matchings with a particular vacancy for the rectangle case.
Paper
A sharp Schrödinger maximal estimate in ℝ
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.
Journal Article