Rank of a partition

In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.

Definition

By a partition of a positive integer n we mean a finite multiset λ = { λ_k, λ_{k − 1}, . . . , λ₁ } of positive integers satisfying the following two conditions:

Rank (formation)

A Rank is a line of military personnel, drawn up in line abreast (i.e. standing side by side).

Usage

Commonly, troops called to 'By the right, fall in!' do so by forming in line abreast, determining their initial position in relation to a marker. This may be a position on the ground or a single person placed previously to the movement. Often troops determine their spacing on the rank by extending their right arm to touch the left shoulder of the marker, or the person on that marked position, and, then starting from the right, align themselves visually to form a straight row. The alignment may then be checked by a non-commissioned officer making observations and calling orders from the end of the rank. (The order may be given to fall in by the left).

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Rank (film)

Rank is a 2002 fourteen-minute short film directed by David Yates. It was nominated for the British Academy Film Award for Best Short Film at the BAFTAs.

Development

Producer Andrew O'Connell said "We first went to the London Production Fund who gave us the full grant of £15,000. Then we approached Scottish Screen but as we were not a Scots production company they could only give us £5,000. We got Scottish co-producers, re-applied and got £25,000. We also got a lot of deals on post production and post production finance from The First Film Foundation – who are backed by UGC and Universal".

In an interview with Film London, director David Yates stated that he "wanted to use non actors to tell the story, to create a reality. It was also a big break for our writer, Robbie – because it was essentially his first film and for all the kids we cast in Glasgow who had never done a film before." He went on to say that he had "just finished a period drama called The Way We Live Now, which had taken me a year, and which was a very big production ... but fairly formal in many ways as a piece of work ... this [Rank] was an opportunity to just shake all of that off and get back to my roots."

Angel (season 5)

The fifth and final season of the television series Angel, the spin-off of Buffy the Vampire Slayer, premiered on October 1, 2003 on The WB and concluded its 22-episode season and its television run on May 19, 2004. The season aired on Wednesdays at 9:00 pm ET. This was the first and only season of Angel to air following the finale of Buffy the Vampire Slayer.

Season synopsis

Because of You (Ne-Yo album)

Because of You is the second studio album by American recording artist Ne-Yo, released on April 25, 2007 by Def Jam Recordings in the United States. It features guest contributions from rapper Jay-Z on "Crazy" and singer Jennifer Hudson on "Leaving Tonight". Because of You debuted at number one on the Billboard 200 and Top R&B/Hip-Hop Albums charts, selling over 250,000 copies in its first week. Upon its release, the album received generally positive reviews from most music critics, based on an aggregate score of 74/100 from Metacritic. It won a Grammy Award for Best Contemporary R&B Album at the 2008 Grammy Awards.

Critical reception

The album has drawn mainly positive reviews, with Entertainment Weekly magazine noting, "Indeed, the album is an unmistakable attempt to channel [Michael Jackson's early work such as "Off the Wall"]...and the effort often pays off beautifully...Ne-Yo's lithe falsetto puts the many others who've been labeled Jackson-esque to shame." PopMatters.com called the album "a masterpiece of a record" and praised the title track's blend of "upbeat modern-day funk tempo, beautifully melodic backdrop and love-sick (in a good way) vocals...[it's] the perfect lead single off an album that is filled with number-one hits".Slant Magazine also compared the album's sensual ballads to Janet Jackson.

Angel (Theory of a Deadman song)

Angel is the third single on Theory of a Deadman's fifth studio album Savages. The single was released on February 24, 2015.

Composition

"Angel" is a ballad about a man who's in love with an angel but realizes that he eventually has to let her go. Randy Shatkowski of Underground Pulse describes the song as an "electronic-tinged lost love ballad" and noted Tyler Connolly's vocals to be his most vulnerable yet.

Chart positions

"Angel" peaked at No. 2 on the Mainstream Rock chart, making it the band's highest peaking single there since "Lowlife" reached No. 1 in 2011. The song has also gained airplay on SiriusXM the Pulse.

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