Ulam spiral

Independent set (graph theory) Unrelated vertices in graphs of vertices such that for every two vertices in A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph . and is usually denoted by Every maximum independent set also is maximal, but the converse implication does not necessarily hold. Properties[edit] Relationship to other graph parameters[edit] and the size of a minimum vertex cover is equal to the number of vertices in the graph. A vertex coloring of a graph corresponds to a partition of its vertex set into independent subsets. , is at least the quotient of the number of vertices in and the independent number Maximal independent set[edit] Finding independent sets[edit] In computer science, several computational problems related to independent sets have been studied. Maximum independent sets and maximum cliques[edit] Exact algorithms[edit] Approximation algorithms[edit] See also[edit]

Ant on a rubber rope Mathematics problem The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary,[1][2] but a typical form is as follows: An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). An ant (red dot) crawling on a stretchable rope at a constant speed of 1 cm/s. A formal statement of the problem [edit] For sake of analysis, the following is a formalized version of the puzzle. Consider an ideal elastic rope on the -axis such that at time its endpoints are at (the starting point) and (the target point) for constants and . the target point is at the position and that as varies the target point moves at constant velocity . it begins at the starting point, moving along the rope at constant velocity is 1 km, is 1 km/s, and is 1 cm/s.

linear algebra - What is the difference between a point and a vector? 1 + 2 + 3 + 4 + ⋯ The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is −1/12. The sum of all natural numbers 1 + 2 + 3 + 4 + · · · is a divergent series. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory. In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".[2] Partial sums[edit] The first six triangular numbers The partial sums of the series 1 + 2 + 3 + 4 + 5 + ⋯ are 1, 3, 6, 10, 15, etc. This equation was known to the Pythagoreans as early as the sixth century B.C.E.[3] Numbers of this form are called triangular numbers, because they can be arranged in a triangle. Summability[edit] Heuristics[edit] Dividing both sides by −3, one gets c = −1/12. . . .