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.

Things (application)

Things is an award-winning task management app for OS X, iOS, and watchOS made by Cultured Code, a software startup based in Stuttgart, Germany. It was the first modern task manager available for Mac when it released to 12,000 people as an alpha in late 2007, and quickly gained popularity. The following summer, when the App Store launched, it was among the first 500 apps available for iPhone. It was then released alongside the iPad in April 2010, and became one of the first apps available for Apple Watch in March 2015.

In December 2013, Cultured Code announced that they had sold one million copies of the software to date, and in December 2014 the company announced that downloads had increased by an additional three million.

Awards

Things has won multiple awards over the years. It first won the MacLife Editors' Choice Award in 2008, and then in 2009 it went on to win the Apple Design Award, the Macworld Editors’ Choice Award, and the Macworld Best of Show Award. In 2012, after the release of Things 2, Apple selected it as Editors’ Choice, named it among the App Store Best of 2012, and inducted it into the App Store Hall of Fame.

Thing

Thing or The Thing may refer to:

Philosophy

History

Film and television

Comics

Video games

Things (Bobby Darin song)

"Things" is a song which was written and recorded by Bobby Darin in 1962. It reached #3 in the US, #2 in the UK, and #3 in the first ever official Irish Singles Chart, published by RTÉ in October 1962. However, there had been an Irish Top 10 before 1962, published in the Evening Herald from February 1959. As a result, "Things" actually peaked at #2 in the "unofficial" Evening Herald Chart.

In 1962, Darin began to write and sing country music, with hit songs including "Things". It was the final Darin single released on the Atco Records unit of Atlantic Records before he began recording for Capitol Records. While vault material would continue to be issued on Atco, Darin would later return to Atlantic Records. The song was sung by Dean Martin and Nancy Sinatra in the 1967 TV special Movin' with Nancy, starring Nancy Sinatra, which was released to home video in 2000.

A cover of the song by Anne Murray from her 1976 album Keeping in Touch peaked at #12 on the adult contemporary chart.