Determinacy

In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.

Basic notions

Games

The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers.

In this sort of game we consider two players, often named I and II, who take turns playing natural numbers, with I going first. They play "forever"; that is, their plays are indexed by the natural numbers. When they're finished, a predetermined condition decides which player won. This condition need not be specified by any definable rule; it may simply be an arbitrary (infinitely long) lookup table saying who has won given a particular sequence of plays.

More formally, consider a subset A of Baire space; recall that the latter consists of all ω-sequences of natural numbers. Then in the game G_A, I plays a natural number a₀, then II plays a₁, then I plays a₂, and so on. Then I wins the game if and only if

And I

"And I" is a song written by American R&B singer Ciara, and produced by Adonis Shropshire for Ciara's first album, Goodies (2004). It was released as the album's fourth and final single on August 30, 2005 (see 2005 in music) in North America.

The song was originally the third choice for the fourth single. Ciara has mentioned in an interview that "Thug Style", "Hotline" or "Pick up the Phone" would be the fourth single, but when Ciara suggested "And I" to executives at her previous record label LaFace, they were unhappy because they wanted "Next to You" to be the fourth single.

Chart performance

At the time of the U.S. release of "And I", Ciara's collaboration with Bow Wow, "Like You", and her collaboration with Missy Elliott, "Lose Control", were at the height of their popularity. "And I" peaked at 96 on the U.S. Billboard Hot 100 and 27 on Billboard's Hot R&B/Hip-Hop Singles & Tracks chart, which were poor positions compared to those attained by the first three singles from the album Goodies. It was not released outside North America because of this, and it is Ciara's least successful single.

Love (disambiguation)

Love is an emotion of strong affection and personal attachment.

Love or Loved may also refer to:

In arts and entertainment

In film and television

Music video

A music video or song video is a short film integrating a song and imagery, produced for promotional or artistic purposes. Modern music videos are primarily made and used as a marketing device intended to promote the sale of music recordings. There are also cases where songs are used in tie in marketing campaigns that allow them to become more than just a song. Tie ins and merchandising could be used in toys are marketing campaigns for food and other products. Although the origins of music videos date back to musical short films that first appeared in the 1920s, they came into prominence in the 1980s when MTV based their format around the medium. Prior to the 1980s, these works were described by various terms including "illustrated song", "filmed insert", "promotional (promo) film", "promotional clip", "promotional video", "song video", "song clip" or "film clip". Since the creation and increased popularity of YouTube, mainstream artists now promote new music videos by releasing trailers of short promos on the site for their upcoming song and music video. Consequentially, YouTube has been converted into a social media platform for celebrities and artists to market themselves to their fans and audiences.