Periodic sequence
In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over:
The number p of repeated terms is called the period (period).
Definition
A periodic sequence is a sequence a1, a2, a3, ... satisfying
for all values of n. If we regard a sequence as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.
Examples
The sequence of digits in the decimal expansion of 1/7 is periodic with period six:
More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).
The sequence of powers of −1 is periodic with period two:
More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.
A periodic point for a function ƒ: X → X is a point p whose orbit
is a periodic sequence. Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.
Cycle (film)
Cycle is a 2008 Indian Malayalam film directed by Johny Antony and written by James Albert, starring Vineeth Sreenivasan, Vinu Mohan, Sandhya, Bhama, and Jagathy Sreekumar. The tale is about two friends, Roy (Vineeth Sreenivasan) and Sanju (Vinu Mohan), struggling to make both ends meet with their low salaries, working respectively as a cashier in a private finance company and as a salesman in an electronics goods shop. It was a box office success.
Synopsis
The film, set at City Centre, a shopping mall in Kochi as backdrop, is all about a few youngsters, Roy and Sanju and their love interests, Annie and Meenakshy. Long-ignored and mistreated by society for reasons none of theirs, they decide to take matters into their own hands. Roy (Vineeth Sreenivasan) is an accountant in a private financing company of Kaustubhan, (deftly played with ease and characteristic gags by Jagathy) who pays his staff, the modest of all salaries. Sanju (Vinu Mohan) is a desperate cricketer, left out of Ranji team, who is working in the same commercial mall in an electronics shop. Roy has a soft corner for Annie (Bhama) who works as a receptionist in a travel agency in the same complex. And Meenakshy (Sandhya), the daughter of Kaushtubhan, is madly in love, always chasing Sanju.
Cycle (Paul Horn album)
Cycle is an album by Paul Horn which was originally released on the RCA Victor label in 1965.
Reception
The Allmusic site awarded the album 2 stars stating: "One can hear hints of Paul Horn's future directions on this obscure LP. ...so this is not an album for everyone."Cycle, credited to The Paul Horn Quintet, was nonetheless nominated for a 1965 Grammy Award for Best Jazz Album, Small Ensemble.
Within three years, Horn would abandon jazz altogether to work on atmospheric mood music".
Track listing
All compositions by Paul Horn except as indicated
Personnel
Cycle (magazine)
Cycle Magazine was an American motorcycling enthusiast magazine, published from the early 1950s through the early 1990s. During its heyday, in the 1970s and 1980s, it had a circulation of more than 500,000 and was headquartered in Westlake Village, California, near the canyon roads of the Santa Monica Mountains, where Cycle's editors frequently road tested and photographed test bikes.
History
Cycle was founded by Robert E. Petersen of Trend Inc. and Petersen Publishing, which also published Hot Rod and Motor Trend magazines. Petersen sold Cycle to Floyd Clymer in July 1953. Floyd Clymer, a pioneer in the sport of motorcycling, was a racer, a motorcycle dealer and distributor, a magazine publisher, a racing promoter, an author, and a motorcycle manufacturer. He was inducted into the Motorcycle Hall of Fame in 1998. In an anniversary issue of Cycle, his editorial approach was summed up as: "Clymer never met a motorcycle he didn't like. Clymer owned Cycle until 1966, when he sold the publication to the New York-based publishing company Ziff-Davis Publications, which owned it through the mid-1980s. CBS, which also owned Cycle's main competitor, Cycle World, purchased Cycle in 1985; Diamandis Communications owned both magazines for a short time in 1988. In April of that year both were sold to Hachette Filipacchi Media U.S.
Cycle (graph theory)
In graph theory, there are several different types of object called cycles, principally a closed walk and a simple cycle; also, e.g., an element of the cycle space of the graph.
A closed walk consists of a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph. In a directed graph, each edge must be traversed by the walk consistently with its direction: the edge must be oriented from the earlier of two consecutive vertices to the later of the two vertices in the sequence. The choice of starting vertex is not important: traversing the same cyclic sequence of edges from different starting vertices produces the same closed walk.
A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex, or as the set of edges in such a walk. The two definitions are equivalent in directed graphs, where simple cycles are also called directed cycles: the cyclic sequence of vertices and edges in a walk is completely determined by the set of edges that it uses. In undirected graphs the set of edges of a cycle can be traversed by a walk in either of two directions, giving two possible directed cycles for every undirected cycle. (For closed walks more generally, in directed or undirected graphs, the multiset of edges does not unambiguously determine the vertex ordering.) A circuit can be a closed walk allowing repetitions of vertices but not edges; however, it can also be a simple cycle, so explicit definition is recommended when it is used.
Cycle (album)
Cycle is an album by the Japanese noise musician Merzbow. It features recordings of Masami Akita's pet chickens. Material from this album was performed live at Radio Nova in Oslo, which was released as Mini Cycle / Yoshino Tamago / Yonos Bigfoot.
Track listing
All music composed by Masami Akita.
Personnel
References
Podcasts:
