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As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction 5 / 7 when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2 ...
The algorithm was first published in 1944 by Kenneth Levenberg, [1] while working at the Frankford Army Arsenal. It was rediscovered in 1963 by Donald Marquardt, [2] who worked as a statistician at DuPont, and independently by Girard, [3] Wynne [4] and Morrison. [5] The LMA is used in many software applications for solving generic curve-fitting ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
A most-perfect magic square of order n is a magic square containing the numbers 1 to n 2 with two additional properties: Each 2 × 2 subsquare sums to 2s, where s = n 2 + 1. All pairs of integers distant n/2 along a (major) diagonal sum to s.
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers : if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length.
Chess is an example of a game of perfect information.. In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition.
A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with = 4/3 F, = 1 Ω, = 3/2 H, and = 1/2 H. [3] Taking the impedance of the capacitors to be / and the impedance of the inductors to be , where = + is the complex frequency, the circuit equations yield the transfer function for this device: