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If q = 2, then (a-b) 2 = 2 and there is no integral solution a, b. When q > 2, the equation x 2 + y 2 − qxy − q = 0 defines a hyperbola H and (a,b) represents an integral lattice point on H. If (x,x) is an integral lattice point on H with x > 0, then (since q is integral) one can see that x = 1. This proposition's statement is then true for ...
An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion ), the search path of a foraging animal, or the price of a fluctuating ...
The first volume introduces combinatorial game theory and its foundation in the surreal numbers; partizan and impartial games; Sprague–Grundy theory and misère games. The second volume applies the theorems of the first volume to many games, including nim , sprouts , dots and boxes , Sylver coinage , philosopher's phutball , fox and geese .
The problem was first posed by Henry Dudeney in 1900, as a puzzle in recreational mathematics, phrased in terms of placing the 16 pawns of a chessboard onto the board so that no three are in a line. [2] This is exactly the no-three-in-line problem, for the case =. [3] In a later version of the puzzle, Dudeney modified the problem, making its ...
The term ‖ ‖ = # {: +} penalizes the number of jumps and the term ‖ ‖ = = | | measures fidelity to data x. The parameter γ > 0 controls the tradeoff between regularity and data fidelity . Since the minimizer u ∗ {\displaystyle u^{*}} is piecewise constant the steps are given by the non-zero locations of the gradient ∇ u ∗ ...
The numbers on each node indicate the number of possible moves that can be made from that position. The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem.
Rosetta Code is a wiki-based programming chrestomathy website with implementations of common algorithms and solutions to various programming problems in many different programming languages. [ 1 ] [ 2 ] It is named for the Rosetta Stone , which has the same text inscribed on it in three languages, and thus allowed Egyptian hieroglyphs to be ...
Mark Pilgrim is a software developer, writer, and advocate of free software.He authored a popular blog, and has written several books, including Dive into Python, a guide to the Python programming language published under the GNU Free Documentation License.