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 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 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