Function

Function may refer to:

Math, computing and engineering

Music

Other uses

See also

Minkowski's question mark function

In mathematics, the Minkowski question mark function (or the slippery devil's staircase), denoted by ?(x), is a function possessing various unusual fractal properties, defined by Hermann Minkowski (1904, pages 171–172). It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

Definition

If is the continued fraction representation of an irrational number x, then

whereas:

If is a continued fraction representation of a rational number x, then

Intuitive explanation

To get some intuition for the definition above, consider the different ways of interpreting an infinite string of bits beginning with 0 as a real number in [0,1]. One obvious way to interpret such a string is to place a binary point after the first 0 and read the string as a binary expansion: thus, for instance, the string 001001001001001001001001... represents the binary number 0.010010010010..., or 2/7. Another interpretation views a string as the continued fraction [0; a₁, a₂, … ], where the integers a_i are the run lengths in a run-length encoding of the string. The same example string 001001001001001001001001... then corresponds to [0; 2, 1, 2, 1, 2, 1, …] = √3 − 1/2. If the string ends in an infinitely long run of the same bit, we ignore it and terminate the representation; this is suggested by the formal "identity":

Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x². The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Electricity

Electricity is the set of physical phenomena associated with the presence and flow of electric charge. Electricity gives a wide variety of well-known effects, such as lightning, static electricity, electromagnetic induction and electric current. In addition, electricity permits the creation and reception of electromagnetic radiation such as radio waves.

In electricity, charges produce electromagnetic fields which act on other charges. Electricity occurs due to several types of physics:

Electric (Pet Shop Boys album)

Electric is the twelfth studio album by English synthpop duo Pet Shop Boys, released on 12 July 2013, and is their first album not to be released on Parlophone. It was released on the duo's own label, x2, through Kobalt Label Services.

In their native United Kingdom, Electric was BBC Radio 2's Album of the Week from 8 July 2013. It features a collaboration with British singer-songwriter/rapper Example.

Electric hit No. 3 and No. 26 in the UK and US respectively, their highest chart performances in both territories in 20 years. The album also performed very well in other markets. It was supported by the Electric Tour which visited 46 countries through 112 concerts.

Background and release

Pet Shop Boys recorded the album in London, Berlin and Los Angeles from November 2012 to April 2013. On 30 April 2013, the duo announced on their website that they had recorded the album over the "last six months", with "Fluorescent" in particular "written and recorded within the last month". Stuart Price was chosen to produce the album. Price finished mixing the album in late April 2013.

Electric (Robyn song)

"Electric" is a song by Swedish recording artist Robyn from her second studio album My Truth (1999). It was released as the album's lead single on 29 April 1999 by BMG Sweden. Robyn wrote the track in collaboration with its producers Ulf Lindström and Johan Ekhé. The single artwork features the singer wearing a feather headpiece designed by Sebastian Wahl. Musically, "Electric" is an electronic funk song, and the lyrics address unexpected life events that make one feel alive.

"Electric" was hailed by music critics as a highlight on My Truth. It achieved commercial success in Sweden, with a peak position of number six on the Sverigetopplistan chart and a gold certification by the Swedish Recording Industry Association (GLF). Though it was initially planned to be serviced internationally, "Electric" and its parent album were never released outside of Sweden due to a dispute between Robyn and her overseas label RCA Records.

Background

"Electric" was written by Robyn, Ulf Lindström and Johan Ekhé. Lindström and Ekhé recorded Robyn's vocals and produced the track at Lifeline Studios in Stockholm, Sweden. Lindström and Patrik Berger played the guitar, while all other instruments as well as mixing were handled by Lindström and Ekhé. Hernan "Boogie" Donoso served as an assistant engineer and Björn Engelmann mastered the song at Cutting Room Studios in Stockholm. "Electric" is an electronic funk song with elements of R&B and reggae music. In a chat room provided by Swedish tabloid newspaper Aftonbladet, Robyn elaborated on the lyrical matter of the song when asked by a fan; she stated that it is about unexpected events in life, both good and bad, that are "fantastic" and make one feel alive.