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The optimal packing of 15 circles in a square Optimal solutions have been proven for n ≤ 30. Packing circles in a rectangle; Packing circles in an isosceles right triangle - good estimates are known for n < 300. Packing circles in an equilateral triangle - Optimal solutions are known for n < 13, and conjectures are available for n < 28. [14]
Happily the context of 51 and 52, together with the base, mid-line, and smaller triangle area (which are given as 4 + 1/2, 2 + 1/4 and 7 + 1/2 + 1/4 + 1/8, respectively) make it possible to interpret the problem and its solution as has been done here. The given paraphrase therefore represents a consistent best guess as to the problem's intent ...
The golden ratio's negative −φ and reciprocal φ−1 are the two roots of the quadratic polynomial x2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. This quadratic polynomial has two roots, and. The golden ratio is also closely related to the polynomial.
Solutions (not necessarily optimal) have been computed for every N ≤ 10,000. [2] Solutions up to N = 20 are shown below. [2] The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest square numbers), but ceases to be optimal for larger squares from 49 onwards.
For instance, problem 19 asks one to calculate a quantity taken 1 + 1 ⁄ 2 times and added to 4 to make 10. [1] In other words, in modern mathematical notation one is asked to solve 3 2 x + 4 = 10 {\displaystyle {\frac {3}{2}}x+4=10} .
[2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6] Additional proofs were published in the 17th century by Pietro Mengoli [2] [7] and by Jacob Bernoulli.
QR decomposition. In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular ...
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [ 1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [ 2] Since the problem had withstood the attacks ...